Which Expression Is Equivalent To $4^{-2}$?A. 1 2 × 1 2 × 1 2 × 1 2 \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} 2 1 × 2 1 × 2 1 × 2 1 B. ( 4 ) ( − 2 (4)(-2 ( 4 ) ( − 2 ] C. 1 4 × 1 4 \frac{1}{4} \times \frac{1}{4} 4 1 × 4 1 D. ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 (-2)(-2)(-2)(-2 ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ]

Mar 11, 2025 by ADMIN 345 views

**Understanding Exponents and Equivalent Expressions**

Introduction

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will explore the concept of exponents and equivalent expressions, focusing on the expression $4^{-2}$ . We will examine the given options and determine which one is equivalent to this expression.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication of a number. For example, $2^3$ can be read as "2 to the power of 3" and is equivalent to $2 \times 2 \times 2$ . Exponents can be positive, negative, or zero.

Negative Exponents

A negative exponent is a fraction with the exponent in the denominator. For example, $2^{-3}$ is equivalent to $\frac{1}{2^3}$ . This means that we can rewrite a negative exponent as a fraction with the base in the denominator.

Equivalent Expressions

An equivalent expression is a mathematical statement that has the same value as another expression. In the case of exponents, equivalent expressions can be obtained by applying the rules of exponents.

Option A: $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}$

Option A represents the product of four fractions, each with a denominator of 2. This can be rewritten as $\frac{1}{2^4}$ , which is equivalent to $2^{-4}$ .

Option B: $(4)(-2)$

Option B represents the product of two numbers, 4 and -2. However, this expression does not involve exponents and is not equivalent to $4^{-2}$ .

Option C: $\frac{1}{4} \times \frac{1}{4}$

Option C represents the product of two fractions, each with a denominator of 4. This can be rewritten as $\frac{1}{4^2}$ , which is equivalent to $4^{-2}$ .

Option D: $(-2)(-2)(-2)(-2)$

Option D represents the product of four numbers, each with a value of -2. However, this expression does not involve exponents and is not equivalent to $4^{-2}$ .

Conclusion

In conclusion, the correct answer is Option C: $\frac{1}{4} \times \frac{1}{4}$ . This expression is equivalent to $4^{-2}$ , as it represents the product of two fractions with a denominator of 4. The other options do not involve exponents and are not equivalent to $4^{-2}$ .

Key Takeaways

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number.
Negative exponents can be rewritten as fractions with the base in the denominator.
Equivalent expressions can be obtained by applying the rules of exponents.
The correct answer is Option C: $\frac{1}{4} \times \frac{1}{4}$ , which is equivalent to $4^{-2}$ .

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Mathematics for the Nonmathematician" by Morris Kline
[3] "The Art of Mathematics" by Bela Bollobas
Exponents and Equivalent Expressions: Q&A =============================================

Introduction

In our previous article, we explored the concept of exponents and equivalent expressions, focusing on the expression $4^{-2}$ . We examined the given options and determined which one is equivalent to this expression. In this article, we will provide a Q&A section to help clarify any doubts and provide further understanding of the topic.

Q: What is an exponent?

A: An exponent is a shorthand way of representing repeated multiplication of a number. For example, $2^3$ can be read as "2 to the power of 3" and is equivalent to $2 \times 2 \times 2$ .

Q: What is a negative exponent?

A: A negative exponent is a fraction with the exponent in the denominator. For example, $2^{-3}$ is equivalent to $\frac{1}{2^3}$ . This means that we can rewrite a negative exponent as a fraction with the base in the denominator.

Q: How do I rewrite a negative exponent as a fraction?

A: To rewrite a negative exponent as a fraction, simply place the base in the denominator and make the exponent positive. For example, $2^{-3}$ can be rewritten as $\frac{1}{2^3}$ .

Q: What is an equivalent expression?

A: An equivalent expression is a mathematical statement that has the same value as another expression. In the case of exponents, equivalent expressions can be obtained by applying the rules of exponents.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you can apply the rules of exponents and simplify both expressions. If the simplified expressions are the same, then the original expressions are equivalent.

Q: What is the correct answer for the expression $4^{-2}$ ?

A: The correct answer is Option C: $\frac{1}{4} \times \frac{1}{4}$ . This expression is equivalent to $4^{-2}$ , as it represents the product of two fractions with a denominator of 4.

Q: Can you provide more examples of equivalent expressions?

A: Here are a few more examples:

$2^3$ is equivalent to $2 \times 2 \times 2$
$3^{-2}$ is equivalent to $\frac{1}{3^2}$
$5^4$ is equivalent to $5 \times 5 \times 5 \times 5$

Q: How do I apply the rules of exponents to simplify expressions?

A: To apply the rules of exponents, you can follow these steps:

Simplify any negative exponents by rewriting them as fractions.
Apply the product rule: $a^m \times a^n = a^{m+n}$
Apply the power rule: $(a^m)^n = a^{m \times n}$
Simplify any remaining exponents.

Conclusion

In conclusion, we hope this Q&A section has helped clarify any doubts and provided further understanding of the topic of exponents and equivalent expressions. Remember to always apply the rules of exponents and simplify expressions to determine if they are equivalent.

Key Takeaways

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number.
Negative exponents can be rewritten as fractions with the base in the denominator.
Equivalent expressions can be obtained by applying the rules of exponents.
The correct answer for the expression $4^{-2}$ is Option C: $\frac{1}{4} \times \frac{1}{4}$ .

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Mathematics for the Nonmathematician" by Morris Kline
[3] "The Art of Mathematics" by Bela Bollobas

Introduction

What are Exponents?

Negative Exponents

Equivalent Expressions

Option A: $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}$

Option B: $(4)(-2)$

Option C: $\frac{1}{4} \times \frac{1}{4}$

Option D: $(-2)(-2)(-2)(-2)$

Conclusion

Key Takeaways

Further Reading

References

Introduction

Q: What is an exponent?

Q: What is a negative exponent?

Q: How do I rewrite a negative exponent as a fraction?

Q: What is an equivalent expression?

Q: How do I determine if two expressions are equivalent?

Q: What is the correct answer for the expression $4^{-2}$ ?

Q: Can you provide more examples of equivalent expressions?

Q: How do I apply the rules of exponents to simplify expressions?

Conclusion

Key Takeaways

Further Reading

References

Introduction

What are Exponents?

Negative Exponents

Equivalent Expressions

Option A: 12×12×12×12\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}21​×21​×21​×21​

Option B: (4)(−2)(4)(-2)(4)(−2)

Option C: 14×14\frac{1}{4} \times \frac{1}{4}41​×41​

Option D: (−2)(−2)(−2)(−2)(-2)(-2)(-2)(-2)(−2)(−2)(−2)(−2)

Conclusion

Key Takeaways

Further Reading

References

Introduction

Q: What is an exponent?

Q: What is a negative exponent?

Q: How do I rewrite a negative exponent as a fraction?

Q: What is an equivalent expression?

Q: How do I determine if two expressions are equivalent?

Q: What is the correct answer for the expression 4−24^{-2}4−2?

Q: Can you provide more examples of equivalent expressions?

Q: How do I apply the rules of exponents to simplify expressions?

Conclusion

Key Takeaways

Further Reading

References

Option A: $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}$

Option B: $(4)(-2)$

Option C: $\frac{1}{4} \times \frac{1}{4}$

Option D: $(-2)(-2)(-2)(-2)$

Q: What is the correct answer for the expression $4^{-2}$ ?