Which Expression Is Equivalent To $\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2$?A. $\sqrt[4]{2^3}$B. $\sqrt{2^5}$C. $\sqrt[4]{4^3}$

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Understanding Exponents and Their Properties

When dealing with exponents, it's essential to understand the properties that govern their behavior. One of the most crucial properties is the rule for multiplying exponential expressions with the same base. This rule states that when we multiply two exponential expressions with the same base, we add their exponents. In mathematical terms, this can be represented as:

am⋅an=am+na^m \cdot a^n = a^{m+n}

This property is fundamental in simplifying complex exponential expressions and is a crucial tool in solving various mathematical problems.

Simplifying the Given Expression

The given expression is (212⋅234)2\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2. To simplify this expression, we can start by applying the property of multiplying exponential expressions with the same base. Since both terms in the expression have the same base, which is 2, we can add their exponents.

(212⋅234)2=(212+34)2\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2 = \left(2^{\frac{1}{2} + \frac{3}{4}}\right)^2

Evaluating the Exponent

Now, let's evaluate the exponent by adding the fractions.

12+34=24+34=54\frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4}

So, the expression becomes:

(254)2\left(2^{\frac{5}{4}}\right)^2

Applying the Power Rule

The next step is to apply the power rule, which states that when we raise an exponential expression to a power, we multiply the exponents. In mathematical terms, this can be represented as:

(am)n=am⋅n\left(a^m\right)^n = a^{m \cdot n}

Applying this rule to our expression, we get:

(254)2=254⋅2\left(2^{\frac{5}{4}}\right)^2 = 2^{\frac{5}{4} \cdot 2}

Evaluating the Exponent

Now, let's evaluate the exponent by multiplying the fractions.

54⋅2=5⋅24=104=52\frac{5}{4} \cdot 2 = \frac{5 \cdot 2}{4} = \frac{10}{4} = \frac{5}{2}

So, the expression becomes:

2522^{\frac{5}{2}}

Simplifying the Expression

The expression 2522^{\frac{5}{2}} can be simplified further by expressing the exponent as a product of two fractions.

252=252⋅22=21042^{\frac{5}{2}} = 2^{\frac{5}{2} \cdot \frac{2}{2}} = 2^{\frac{10}{4}}

Evaluating the Exponent

Now, let's evaluate the exponent by simplifying the fraction.

104=52\frac{10}{4} = \frac{5}{2}

So, the expression becomes:

2522^{\frac{5}{2}}

Converting to Radical Form

The expression 2522^{\frac{5}{2}} can be converted to radical form by expressing the exponent as a power of 2.

252=252^{\frac{5}{2}} = \sqrt{2^5}

Conclusion

In conclusion, the expression (212⋅234)2\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2 is equivalent to 25\sqrt{2^5}. This result was obtained by applying the properties of exponents, including the rule for multiplying exponential expressions with the same base and the power rule.

Comparison with Other Options

Let's compare our result with the other options provided.

A. 234\sqrt[4]{2^3}

This option is not equivalent to our result. The expression 234\sqrt[4]{2^3} can be simplified as:

234=2⋅2⋅24=234=234\sqrt[4]{2^3} = \sqrt[4]{2 \cdot 2 \cdot 2} = \sqrt[4]{2^3} = 2^{\frac{3}{4}}

B. 434\sqrt[4]{4^3}

This option is not equivalent to our result. The expression 434\sqrt[4]{4^3} can be simplified as:

434=4⋅4⋅44=434=434=232\sqrt[4]{4^3} = \sqrt[4]{4 \cdot 4 \cdot 4} = \sqrt[4]{4^3} = 4^{\frac{3}{4}} = 2^{\frac{3}{2}}

C. 434\sqrt[4]{4^3}

This option is not equivalent to our result. The expression 434\sqrt[4]{4^3} can be simplified as:

434=4⋅4⋅44=434=434=232\sqrt[4]{4^3} = \sqrt[4]{4 \cdot 4 \cdot 4} = \sqrt[4]{4^3} = 4^{\frac{3}{4}} = 2^{\frac{3}{2}}

Final Answer

The final answer is 25\boxed{\sqrt{2^5}}.

Frequently Asked Questions

Q: What is the rule for multiplying exponential expressions with the same base?

A: The rule for multiplying exponential expressions with the same base states that when we multiply two exponential expressions with the same base, we add their exponents. In mathematical terms, this can be represented as:

am⋅an=am+na^m \cdot a^n = a^{m+n}

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can start by applying the properties of exponents, including the rule for multiplying exponential expressions with the same base and the power rule. The power rule states that when we raise an exponential expression to a power, we multiply the exponents. In mathematical terms, this can be represented as:

(am)n=am⋅n\left(a^m\right)^n = a^{m \cdot n}

Q: What is the power rule for exponents?

A: The power rule for exponents states that when we raise an exponential expression to a power, we multiply the exponents. In mathematical terms, this can be represented as:

(am)n=am⋅n\left(a^m\right)^n = a^{m \cdot n}

Q: How do I convert an exponential expression to radical form?

A: To convert an exponential expression to radical form, you can express the exponent as a power of the base. For example, the expression 2522^{\frac{5}{2}} can be converted to radical form as:

252=252^{\frac{5}{2}} = \sqrt{2^5}

Q: What is the difference between an exponential expression and a radical expression?

A: An exponential expression is a mathematical expression that involves a base raised to a power, such as 232^3. A radical expression, on the other hand, is a mathematical expression that involves a root of a number, such as 4\sqrt{4}.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can start by looking for perfect squares or other factors that can be simplified. For example, the expression 16\sqrt{16} can be simplified as:

16=4⋅4=4⋅4=2⋅2=4\sqrt{16} = \sqrt{4 \cdot 4} = \sqrt{4} \cdot \sqrt{4} = 2 \cdot 2 = 4

Q: What is the relationship between exponential expressions and radical expressions?

A: Exponential expressions and radical expressions are related in that they can be used to represent the same mathematical concept. For example, the expression 2522^{\frac{5}{2}} can be represented as a radical expression as:

252=252^{\frac{5}{2}} = \sqrt{2^5}

Q: How do I choose between using an exponential expression or a radical expression?

A: When choosing between using an exponential expression or a radical expression, consider the context and the audience. Exponential expressions are often used in mathematical proofs and derivations, while radical expressions are often used in algebra and geometry.

Q: What are some common mistakes to avoid when working with exponents and radicals?

A: Some common mistakes to avoid when working with exponents and radicals include:

  • Forgetting to apply the power rule when raising an exponential expression to a power
  • Forgetting to simplify radical expressions by looking for perfect squares or other factors
  • Confusing the order of operations when working with exponents and radicals

Conclusion

In conclusion, understanding exponents and their properties is essential for working with mathematical expressions. By applying the properties of exponents, including the rule for multiplying exponential expressions with the same base and the power rule, you can simplify complex expressions and convert them to radical form. Remember to choose between using an exponential expression or a radical expression based on the context and the audience, and avoid common mistakes such as forgetting to apply the power rule or simplifying radical expressions.