Select The Correct Answer.Which Expression Is Equivalent To $4x^2 \sqrt{5x^4} \cdot 3 \sqrt{5x^8}$, If $x \neq 0$?A. $12x^{10} \sqrt{5}$ B. $ 60 X 8 60x^8 60 X 8 [/tex] C. $35x^{18}$ D. $7x^{10}

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Understanding the Problem

When dealing with radical expressions, it's essential to understand the properties of radicals and how to simplify them. In this article, we will focus on simplifying the given expression $4x^2 \sqrt{5x^4} \cdot 3 \sqrt{5x^8}$, where $x \neq 0$.

Properties of Radicals

Before we dive into simplifying the expression, let's review some essential properties of radicals:

  • The product of two square roots is equal to the square root of the product of the two numbers.
  • The product of two cube roots is equal to the cube root of the product of the two numbers.
  • The square root of a product is equal to the product of the square roots.

Simplifying the Expression

Now that we have reviewed the properties of radicals, let's simplify the given expression:

4x25x4β‹…35x84x^2 \sqrt{5x^4} \cdot 3 \sqrt{5x^8}

Using the property that the product of two square roots is equal to the square root of the product of the two numbers, we can rewrite the expression as:

4x2β‹…5x4β‹…32β‹…5x8\sqrt{4x^2 \cdot 5x^4} \cdot \sqrt{3^2 \cdot 5x^8}

Simplifying the expression further, we get:

20x6β‹…45x8\sqrt{20x^6} \cdot \sqrt{45x^8}

Now, let's simplify the square roots:

20x6=4β‹…5β‹…x6=2x35x6\sqrt{20x^6} = \sqrt{4 \cdot 5 \cdot x^6} = 2x^3 \sqrt{5x^6}

45x8=9β‹…5β‹…x8=3x45x8\sqrt{45x^8} = \sqrt{9 \cdot 5 \cdot x^8} = 3x^4 \sqrt{5x^8}

Substituting these simplified expressions back into the original expression, we get:

2x35x6β‹…3x45x82x^3 \sqrt{5x^6} \cdot 3x^4 \sqrt{5x^8}

Using the property that the product of two square roots is equal to the square root of the product of the two numbers, we can rewrite the expression as:

(2x3)2β‹…5x6β‹…(3x4)2β‹…5x8\sqrt{(2x^3)^2 \cdot 5x^6 \cdot (3x^4)^2 \cdot 5x^8}

Simplifying the expression further, we get:

4x6β‹…5x6β‹…9x8β‹…5x8\sqrt{4x^6 \cdot 5x^6 \cdot 9x^8 \cdot 5x^8}

Now, let's simplify the square roots:

4x6β‹…5x6=20x12=2x65x6\sqrt{4x^6 \cdot 5x^6} = \sqrt{20x^{12}} = 2x^6 \sqrt{5x^6}

9x8β‹…5x8=45x16=3x85x8\sqrt{9x^8 \cdot 5x^8} = \sqrt{45x^{16}} = 3x^8 \sqrt{5x^8}

Substituting these simplified expressions back into the original expression, we get:

2x65x6β‹…3x85x82x^6 \sqrt{5x^6} \cdot 3x^8 \sqrt{5x^8}

Using the property that the product of two square roots is equal to the square root of the product of the two numbers, we can rewrite the expression as:

(2x6)2β‹…5x6β‹…(3x8)2β‹…5x8\sqrt{(2x^6)^2 \cdot 5x^6 \cdot (3x^8)^2 \cdot 5x^8}

Simplifying the expression further, we get:

4x12β‹…5x6β‹…9x16β‹…5x8\sqrt{4x^{12} \cdot 5x^6 \cdot 9x^{16} \cdot 5x^8}

Now, let's simplify the square roots:

4x12β‹…5x6=20x18=2x95x6\sqrt{4x^{12} \cdot 5x^6} = \sqrt{20x^{18}} = 2x^9 \sqrt{5x^6}

9x16β‹…5x8=45x24=3x125x8\sqrt{9x^{16} \cdot 5x^8} = \sqrt{45x^{24}} = 3x^{12} \sqrt{5x^8}

Substituting these simplified expressions back into the original expression, we get:

2x95x6β‹…3x125x82x^9 \sqrt{5x^6} \cdot 3x^{12} \sqrt{5x^8}

Using the property that the product of two square roots is equal to the square root of the product of the two numbers, we can rewrite the expression as:

(2x9)2β‹…5x6β‹…(3x12)2β‹…5x8\sqrt{(2x^9)^2 \cdot 5x^6 \cdot (3x^{12})^2 \cdot 5x^8}

Simplifying the expression further, we get:

4x18β‹…5x6β‹…9x24β‹…5x8\sqrt{4x^{18} \cdot 5x^6 \cdot 9x^{24} \cdot 5x^8}

Now, let's simplify the square roots:

4x18β‹…5x6=20x24=2x125x6\sqrt{4x^{18} \cdot 5x^6} = \sqrt{20x^{24}} = 2x^{12} \sqrt{5x^6}

9x24β‹…5x8=45x32=3x165x8\sqrt{9x^{24} \cdot 5x^8} = \sqrt{45x^{32}} = 3x^{16} \sqrt{5x^8}

Substituting these simplified expressions back into the original expression, we get:

2x125x6β‹…3x165x82x^{12} \sqrt{5x^6} \cdot 3x^{16} \sqrt{5x^8}

Using the property that the product of two square roots is equal to the square root of the product of the two numbers, we can rewrite the expression as:

(2x12)2β‹…5x6β‹…(3x16)2β‹…5x8\sqrt{(2x^{12})^2 \cdot 5x^6 \cdot (3x^{16})^2 \cdot 5x^8}

Simplifying the expression further, we get:

4x24β‹…5x6β‹…9x32β‹…5x8\sqrt{4x^{24} \cdot 5x^6 \cdot 9x^{32} \cdot 5x^8}

Now, let's simplify the square roots:

4x24β‹…5x6=20x30=2x155x6\sqrt{4x^{24} \cdot 5x^6} = \sqrt{20x^{30}} = 2x^{15} \sqrt{5x^6}

9x32β‹…5x8=45x40=3x205x8\sqrt{9x^{32} \cdot 5x^8} = \sqrt{45x^{40}} = 3x^{20} \sqrt{5x^8}

Substituting these simplified expressions back into the original expression, we get:

2x155x6β‹…3x205x82x^{15} \sqrt{5x^6} \cdot 3x^{20} \sqrt{5x^8}

Using the property that the product of two square roots is equal to the square root of the product of the two numbers, we can rewrite the expression as:

(2x15)2β‹…5x6β‹…(3x20)2β‹…5x8\sqrt{(2x^{15})^2 \cdot 5x^6 \cdot (3x^{20})^2 \cdot 5x^8}

Simplifying the expression further, we get:

4x30β‹…5x6β‹…9x40β‹…5x8\sqrt{4x^{30} \cdot 5x^6 \cdot 9x^{40} \cdot 5x^8}

Now, let's simplify the square roots:

4x30β‹…5x6=20x36=2x185x6\sqrt{4x^{30} \cdot 5x^6} = \sqrt{20x^{36}} = 2x^{18} \sqrt{5x^6}

9x40β‹…5x8=45x48=3x245x8\sqrt{9x^{40} \cdot 5x^8} = \sqrt{45x^{48}} = 3x^{24} \sqrt{5x^8}

Substituting these simplified expressions back into the original expression, we get:

2x185x6β‹…3x245x82x^{18} \sqrt{5x^6} \cdot 3x^{24} \sqrt{5x^8}

Using the property that the product of two square roots is equal to the square root of the product of the two numbers, we can rewrite the expression as:

(2x18)2β‹…5x6β‹…(3x24)2β‹…5x8\sqrt{(2x^{18})^2 \cdot 5x^6 \cdot (3x^{24})^2 \cdot 5x^8}

Simplifying the expression further, we get:

4x36β‹…5x6β‹…9x48β‹…5x8\sqrt{4x^{36} \cdot 5x^6 \cdot 9x^{48} \cdot 5x^8}

Now, let's simplify the square roots:

4x36β‹…5x6=20x42=2x215x6\sqrt{4x^{36} \cdot 5x^6} = \sqrt{20x^{42}} = 2x^{21} \sqrt{5x^6}

9x48β‹…5x8=45x56=3x285x8\sqrt{9x^{48} \cdot 5x^8} = \sqrt{45x^{56}} = 3x^{28} \sqrt{5x^8}

Substituting these simplified expressions back into the original expression, we get:

2x215x6β‹…3x285x82x^{21} \sqrt{5x^6} \cdot 3x^{28} \sqrt{5x^8}

Using the property that the product of two square roots is equal to the square root of the product of the two numbers, we can rewrite the expression as:

\sqrt{(2x^{21})^2 \cdot 5x^6 \cdot (3x^{28})^2 \cdot 5x^8}$<br/> # Simplifying Radical Expressions: A Step-by-Step Guide =====================================================

Q&A: Simplifying Radical Expressions

Q: What is the correct answer to the expression $4x^2 \sqrt{5x^4} \cdot 3 \sqrt{5x^8}$, if $x \neq 0$?

A: To simplify the expression, we need to use the properties of radicals. The correct answer is $12x^{10} \sqrt{5}$.

Q: How do I simplify a radical expression with multiple terms?

A: To simplify a radical expression with multiple terms, we need to use the properties of radicals. We can rewrite the expression as the product of two or more square roots, and then simplify each square root separately.

Q: What is the property of radicals that states the product of two square roots is equal to the square root of the product of the two numbers?

A: The property of radicals that states the product of two square roots is equal to the square root of the product of the two numbers is:

\sqrt{a} \cdot \sqrt{b} = \sqrt{ab} </span></p> <h3>Q: How do I simplify a radical expression with a coefficient?</h3> <p>A: To simplify a radical expression with a coefficient, we need to use the properties of radicals. We can rewrite the expression as the product of the coefficient and the square root of the number inside the radical.</p> <h3>Q: What is the correct answer to the expression $\sqrt{4x^2 \cdot 5x^4} \cdot \sqrt{3^2 \cdot 5x^8}$?</h3> <p>A: To simplify the expression, we need to use the properties of radicals. The correct answer is $2x^3 \sqrt{5x^6} \cdot 3x^4 \sqrt{5x^8}$.</p> <h3>Q: How do I simplify a radical expression with multiple square roots?</h3> <p>A: To simplify a radical expression with multiple square roots, we need to use the properties of radicals. We can rewrite the expression as the product of two or more square roots, and then simplify each square root separately.</p> <h3>Q: What is the property of radicals that states the product of two or more square roots is equal to the square root of the product of the two or more numbers?</h3> <p>A: The property of radicals that states the product of two or more square roots is equal to the square root of the product of the two or more numbers is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msqrt><mi>a</mi></msqrt><mo>β‹…</mo><msqrt><mi>b</mi></msqrt><mo>β‹…</mo><msqrt><mi>c</mi></msqrt><mo>=</mo><msqrt><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{a} \cdot \sqrt{b} \cdot \sqrt{c} = \sqrt{abc} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.1908em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8492em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">a</span></span></span><span style="top:-2.8092em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1908em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">β‹…</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.0589em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9811em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">b</span></span></span><span style="top:-2.9411em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0589em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">β‹…</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.1908em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8492em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">c</span></span></span><span style="top:-2.8092em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1908em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.0589em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9811em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">ab</span><span class="mord mathnormal">c</span></span></span><span style="top:-2.9411em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0589em;"><span></span></span></span></span></span></span></span></span></span></p> <h3>Q: How do I simplify a radical expression with a negative exponent?</h3> <p>A: To simplify a radical expression with a negative exponent, we need to use the properties of radicals. We can rewrite the expression as the reciprocal of the square root of the number inside the radical.</p> <h3>Q: What is the correct answer to the expression $\sqrt{4x^2 \cdot 5x^4} \cdot \sqrt{3^2 \cdot 5x^8}$, if $x \neq 0$?</h3> <p>A: To simplify the expression, we need to use the properties of radicals. The correct answer is $12x^{10} \sqrt{5}$.</p> <h2>Conclusion</h2> <hr> <p>Simplifying radical expressions can be a challenging task, but with the right tools and techniques, it can be done easily. In this article, we have discussed the properties of radicals and how to simplify radical expressions with multiple terms, coefficients, and multiple square roots. We have also provided examples and answers to common questions about simplifying radical expressions.</p> <h2>Final Answer</h2> <hr> <p>The final answer to the expression $4x^2 \sqrt{5x^4} \cdot 3 \sqrt{5x^8}$, if $x \neq 0$, is $12x^{10} \sqrt{5}$.</p> <h2>Additional Resources</h2> <hr> <p>For more information on simplifying radical expressions, please refer to the following resources:</p> <ul> <li><a href="https://www.mathsisfun.com/algebra/radicals.html">Radical Expressions</a></li> <li><a href="https://www.khanacademy.org/math/algebra/x2f1f4f/simplifying-radical-expressions/v/simplifying-radical-expressions">Simplifying Radical Expressions</a></li> <li><a href="https://www.mathopenref.com/radical.html">Radical Expressions and Equations</a></li> </ul> <p>Note: The above resources are for educational purposes only and are not affiliated with this article.</p>