Simplify The Expression Using Properties Of Exponents:$\[ \frac{6 X^{10} Y}{2 X^7 Y^3} \\]A. \[$\frac{3 X^3}{y^2}\$\] B. \[$\frac{4 X^{17}}{y^4}\$\] C. \[$3 X^{17} Y^4\$\] D. \[$\frac{4 Y^2}{x^3}\$\]

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Introduction

In algebra, exponents are a fundamental concept used to represent repeated multiplication of a number. When dealing with expressions containing exponents, it is essential to apply the properties of exponents to simplify them. In this article, we will focus on simplifying the given expression using the properties of exponents.

The Given Expression

The given expression is 6x10y2x7y3\frac{6 x^{10} y}{2 x^7 y^3}. To simplify this expression, we need to apply the properties of exponents.

Properties of Exponents

Before we proceed with simplifying the expression, let's review the properties of exponents:

  • Product of Powers Property: am⋅an=am+na^m \cdot a^n = a^{m+n}
  • Power of a Power Property: (am)n=am⋅n(a^m)^n = a^{m \cdot n}
  • Quotient of Powers Property: aman=am−n\frac{a^m}{a^n} = a^{m-n}
  • Zero Exponent Property: a0=1a^0 = 1

Simplifying the Expression

Now that we have reviewed the properties of exponents, let's simplify the given expression.

Step 1: Factor Out Common Terms

The first step in simplifying the expression is to factor out common terms from the numerator and denominator.

6x10y2x7y3=62⋅x10x7⋅yy3\frac{6 x^{10} y}{2 x^7 y^3} = \frac{6}{2} \cdot \frac{x^{10}}{x^7} \cdot \frac{y}{y^3}

Step 2: Apply the Quotient of Powers Property

Next, we can apply the quotient of powers property to simplify the expression.

62⋅x10x7⋅yy3=3⋅x10−7⋅yy3\frac{6}{2} \cdot \frac{x^{10}}{x^7} \cdot \frac{y}{y^3} = 3 \cdot x^{10-7} \cdot \frac{y}{y^3}

Step 3: Simplify the Exponents

Now, we can simplify the exponents by applying the quotient of powers property.

3⋅x10−7⋅yy3=3⋅x3⋅yy33 \cdot x^{10-7} \cdot \frac{y}{y^3} = 3 \cdot x^3 \cdot \frac{y}{y^3}

Step 4: Cancel Out Common Factors

Finally, we can cancel out common factors between the numerator and denominator.

3⋅x3⋅yy3=3⋅x3⋅1y23 \cdot x^3 \cdot \frac{y}{y^3} = 3 \cdot x^3 \cdot \frac{1}{y^2}

The Final Answer

After simplifying the expression using the properties of exponents, we get:

3x3y2\frac{3 x^3}{y^2}

This is the correct answer.

Comparison with Other Options

Let's compare our answer with the other options:

  • Option A: 3x3y2\frac{3 x^3}{y^2} (Our answer)
  • Option B: 4x17y4\frac{4 x^{17}}{y^4} (Incorrect)
  • Option C: 3x17y43 x^{17} y^4 (Incorrect)
  • Option D: 4y2x3\frac{4 y^2}{x^3} (Incorrect)

Our answer matches with option A.

Conclusion

In this article, we simplified the given expression using the properties of exponents. We reviewed the properties of exponents, factored out common terms, applied the quotient of powers property, simplified the exponents, and canceled out common factors. Our final answer is 3x3y2\frac{3 x^3}{y^2}, which matches with option A.

Frequently Asked Questions

Q: What are the properties of exponents?

A: The properties of exponents are:

  • Product of Powers Property: am⋅an=am+na^m \cdot a^n = a^{m+n}
  • Power of a Power Property: (am)n=am⋅n(a^m)^n = a^{m \cdot n}
  • Quotient of Powers Property: aman=am−n\frac{a^m}{a^n} = a^{m-n}
  • Zero Exponent Property: a0=1a^0 = 1

Q: How do I simplify an expression using the properties of exponents?

A: To simplify an expression using the properties of exponents, follow these steps:

  1. Factor out common terms from the numerator and denominator.
  2. Apply the quotient of powers property to simplify the expression.
  3. Simplify the exponents by applying the quotient of powers property.
  4. Cancel out common factors between the numerator and denominator.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is 3x3y2\frac{3 x^3}{y^2}.

References

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Introduction

In our previous article, we simplified the expression 6x10y2x7y3\frac{6 x^{10} y}{2 x^7 y^3} using the properties of exponents. In this article, we will answer some frequently asked questions related to simplifying expressions using properties of exponents.

Q&A

Q: What are the properties of exponents?

A: The properties of exponents are:

  • Product of Powers Property: am⋅an=am+na^m \cdot a^n = a^{m+n}
  • Power of a Power Property: (am)n=am⋅n(a^m)^n = a^{m \cdot n}
  • Quotient of Powers Property: aman=am−n\frac{a^m}{a^n} = a^{m-n}
  • Zero Exponent Property: a0=1a^0 = 1

Q: How do I simplify an expression using the properties of exponents?

A: To simplify an expression using the properties of exponents, follow these steps:

  1. Factor out common terms from the numerator and denominator.
  2. Apply the quotient of powers property to simplify the expression.
  3. Simplify the exponents by applying the quotient of powers property.
  4. Cancel out common factors between the numerator and denominator.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is 3x3y2\frac{3 x^3}{y^2}.

Q: Can I use the properties of exponents to simplify expressions with negative exponents?

A: Yes, you can use the properties of exponents to simplify expressions with negative exponents. For example, 1x3=x−3\frac{1}{x^3} = x^{-3}.

Q: How do I handle expressions with zero exponents?

A: When an expression has a zero exponent, it is equal to 1. For example, x0=1x^0 = 1.

Q: Can I use the properties of exponents to simplify expressions with fractional exponents?

A: Yes, you can use the properties of exponents to simplify expressions with fractional exponents. For example, x=x1/2\sqrt{x} = x^{1/2}.

Q: How do I simplify expressions with multiple bases?

A: To simplify expressions with multiple bases, you can use the product of powers property. For example, x2⋅y3=(xy)3x^2 \cdot y^3 = (xy)^3.

Examples

Example 1: Simplifying an Expression with Multiple Bases

Simplify the expression (x2⋅y3)⋅(x4⋅y2)(x^2 \cdot y^3) \cdot (x^4 \cdot y^2).

Solution:

(x2⋅y3)⋅(x4⋅y2)=(x2⋅x4)⋅(y3⋅y2)(x^2 \cdot y^3) \cdot (x^4 \cdot y^2) = (x^2 \cdot x^4) \cdot (y^3 \cdot y^2)

Using the product of powers property, we get:

(x2⋅x4)⋅(y3⋅y2)=(x2+4)⋅(y3+2)(x^2 \cdot x^4) \cdot (y^3 \cdot y^2) = (x^2+4) \cdot (y^3+2)

=x6⋅y5= x^6 \cdot y^5

Example 2: Simplifying an Expression with Negative Exponents

Simplify the expression 1x3\frac{1}{x^3}.

Solution:

1x3=x−3\frac{1}{x^3} = x^{-3}

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions using properties of exponents. We reviewed the properties of exponents, provided examples of how to simplify expressions with multiple bases, negative exponents, and fractional exponents.

Frequently Asked Questions

Q: What are some common mistakes to avoid when simplifying expressions using properties of exponents?

A: Some common mistakes to avoid when simplifying expressions using properties of exponents include:

  • Not factoring out common terms from the numerator and denominator.
  • Not applying the quotient of powers property correctly.
  • Not simplifying the exponents correctly.
  • Not canceling out common factors between the numerator and denominator.

Q: How do I know which property of exponents to use when simplifying an expression?

A: To determine which property of exponents to use when simplifying an expression, follow these steps:

  1. Identify the operation being performed (addition, subtraction, multiplication, or division).
  2. Determine which property of exponents corresponds to the operation being performed.
  3. Apply the corresponding property of exponents to simplify the expression.

Q: Can I use the properties of exponents to simplify expressions with variables in the exponent?

A: Yes, you can use the properties of exponents to simplify expressions with variables in the exponent. For example, x2y=(xy)2x^{2y} = (x^y)^2.

References