Simplify The Expression:${ 7 X^0 Y^{-2}\left(2 Y^4\right) }$

Mar 13, 2025 by ADMIN 62 views

$Simplify the Expression: $7 x^0 y^{-2}\left(2 y^4\right)$$

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Introduction

When simplifying algebraic expressions, it's essential to understand the rules of exponents and how to apply them. In this article, we will focus on simplifying the given expression $7 x^0 y^{-2}\left(2 y^4\right)$ using the properties of exponents.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ can be written as $x \cdot x \cdot x$ . When simplifying expressions with exponents, we need to apply the rules of exponents, which include:

Product of Powers Rule: When multiplying two powers with the same base, add the exponents.
Power of a Power Rule: When raising a power to another power, multiply the exponents.
Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1.
Negative Exponent Rule: A negative exponent can be rewritten as a positive exponent by moving the base to the other side of the fraction bar.

Simplifying the Expression

Now that we have a good understanding of the rules of exponents, let's simplify the given expression $7 x^0 y^{-2}\left(2 y^4\right)$ .

Step 1: Apply the Distributive Property

The first step in simplifying the expression is to apply the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac.

7 x^0 y^{-2}\left(2 y^4\right) = 7 x^0 y^{-2} \cdot 2 y^4

Step 2: Simplify the Coefficient

The next step is to simplify the coefficient 7. Since $x^0$ is equal to 1, we can rewrite the expression as:

7 x^0 y^{-2} \cdot 2 y^4 = 7 \cdot 1 \cdot y^{-2} \cdot 2 y^4

Step 3: Apply the Product of Powers Rule

Now that we have simplified the coefficient, we can apply the product of powers rule, which states that when multiplying two powers with the same base, add the exponents.

7 \cdot 1 \cdot y^{-2} \cdot 2 y^4 = 7 \cdot 2 \cdot y^{-2 + 4}

Step 4: Simplify the Exponents

The next step is to simplify the exponents. Since $-2 + 4 = 2$ , we can rewrite the expression as:

7 \cdot 2 \cdot y^{-2 + 4} = 14 y^2

Step 5: Rewrite the Expression with Positive Exponents

Finally, we can rewrite the expression with positive exponents by moving the base to the other side of the fraction bar.

14 y^2 = \frac{14}{y^0} \cdot y^2

Since $y^0$ is equal to 1, we can rewrite the expression as:

\frac{14}{y^0} \cdot y^2 = 14 \cdot 1 \cdot y^2

Therefore, the simplified expression is:

14 y^2

Conclusion

In this article, we simplified the expression $7 x^0 y^{-2}\left(2 y^4\right)$ using the properties of exponents. We applied the distributive property, simplified the coefficient, applied the product of powers rule, simplified the exponents, and finally rewrote the expression with positive exponents. The simplified expression is $14 y^2$ .

Frequently Asked Questions

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1.

Q: What is the negative exponent rule?

A: A negative exponent can be rewritten as a positive exponent by moving the base to the other side of the fraction bar.

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, multiply the exponents when multiplying two powers with the same base.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, apply the rules of exponents, including the product of powers rule, power of a power rule, zero exponent rule, and negative exponent rule.

Final Answer

The final answer is: $\boxed{14 y^2}$

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Introduction

Simplifying algebraic expressions can be a challenging task, especially when dealing with exponents and variables. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two powers with the same base, add the exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$ .

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, multiply the exponents when multiplying two powers with the same base. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$ .

Q: What is the power of a power rule?

A: The power of a power rule states that when raising a power to another power, multiply the exponents. For example, $(x^2)^3 = x^{2 \cdot 3} = x^6$ .

Q: How do I apply the power of a power rule?

A: To apply the power of a power rule, multiply the exponents when raising a power to another power. For example, $(x^2)^3 = x^{2 \cdot 3} = x^6$ .

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. For example, $x^0 = 1$ .

Q: How do I apply the zero exponent rule?

A: To apply the zero exponent rule, replace any non-zero number raised to the power of zero with 1. For example, $x^0 = 1$ .

Q: What is the negative exponent rule?

A: The negative exponent rule states that a negative exponent can be rewritten as a positive exponent by moving the base to the other side of the fraction bar. For example, $x^{-2} = \frac{1}{x^2}$ .

Q: How do I apply the negative exponent rule?

A: To apply the negative exponent rule, move the base to the other side of the fraction bar and change the sign of the exponent. For example, $x^{-2} = \frac{1}{x^2}$ .

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, apply the rules of exponents, including the product of powers rule, power of a power rule, zero exponent rule, and negative exponent rule.

Q: What is the distributive property?

A: The distributive property states that for any real numbers a, b, and c, a(b + c) = ab + ac.

Q: How do I apply the distributive property?

A: To apply the distributive property, multiply each term inside the parentheses by the term outside the parentheses. For example, 2(x + 3) = 2x + 6.

Q: How do I simplify an expression with variables?

A: To simplify an expression with variables, apply the rules of exponents and the distributive property.

Conclusion

Simplifying algebraic expressions can be a challenging task, but by understanding the rules of exponents and the distributive property, you can simplify even the most complex expressions. Remember to apply the product of powers rule, power of a power rule, zero exponent rule, and negative exponent rule to simplify expressions with exponents. Also, don't forget to apply the distributive property to simplify expressions with variables.

Final Answer

The final answer is: $\boxed{14 y^2}$