Select All The Expressions That Are Equivalent To $\left(4^{-1}\right)^{-6}$.A. $4^6$B. $4^{-7}$C. $\frac{1}{4^{-7}}$D. $\frac{1}{4^{-6}}$

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. The expression $\left(4^{-1}\right)^{-6}$ involves both positive and negative exponents, which can be simplified using the rules of exponents. In this article, we will explore the equivalent expressions for $\left(4^{-1}\right)^{-6}$ and discuss the reasoning behind each option.

The Rules of Exponents

Before diving into the equivalent expressions, it's essential to understand the rules of exponents. The power rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, the following equation holds:

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

Additionally, the rule for negative exponents states that for any number $a$ and any integer $n$, the following equation holds:

$$a^{-n} = \frac{1}{a^n}$$

Simplifying the Expression

Using the power rule, we can simplify the expression $\left(4^{-1}\right)^{-6}$ as follows:

$$\left(4^{-1}\right)^{-6} = 4^{-1 \cdot -6} = 4^6$$

This simplification is based on the power rule, which states that the exponent of the inner expression is multiplied by the exponent of the outer expression.

Option A: $4^6$

Option A, $4^6$, is indeed an equivalent expression to $\left(4^{-1}\right)^{-6}$. This is because we simplified the expression using the power rule, resulting in $4^6$.

Option B: $4^{-7}$

Option B, $4^{-7}$, is not an equivalent expression to $\left(4^{-1}\right)^{-6}$. This is because the exponent of $4$ is $-7$, whereas the simplified expression is $4^6$.

Option C: $\frac{1}{4^{-7}}$

Option C, $\frac{1}{4^{-7}}$, is not an equivalent expression to $\left(4^{-1}\right)^{-6}$. This is because the exponent of $4$ is $-7$, whereas the simplified expression is $4^6$.

Option D: $\frac{1}{4^{-6}}$

Option D, $\frac{1}{4^{-6}}$, is not an equivalent expression to $\left(4^{-1}\right)^{-6}$. This is because the exponent of $4$ is $-6$, whereas the simplified expression is $4^6$.

Conclusion

In conclusion, the only equivalent expression to $\left(4^{-1}\right)^{-6}$ is Option A, $4^6$. This is because we simplified the expression using the power rule, resulting in $4^6$. The other options, $4^{-7}$, $\frac{1}{4^{-7}}$, and $\frac{1}{4^{-6}}$, are not equivalent expressions.

Key Takeaways

• The power rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, the following equation holds: $\left(a^m\right)^n = a^{m \cdot n}$.
• The rule for negative exponents states that for any number $a$ and any integer $n$, the following equation holds: $a^{-n} = \frac{1}{a^n}$.
• To simplify the expression $\left(4^{-1}\right)^{-6}$, we can use the power rule, resulting in $4^6$.

Practice Problems

1. Simplify the expression $\left(2^3\right)^{-4}$ using the power rule.
2. Simplify the expression $\left(3^{-2}\right)^{-5}$ using the power rule.
3. Simplify the expression $\left(5^4\right)^{-3}$ using the power rule.

Answer Key

1. $\frac{1}{2^{12}}$
2. $3^{10}$
3. $\frac{1}{5^{12}}$
   Frequently Asked Questions (FAQs) =====================================

In this article, we will address some of the most frequently asked questions related to the expression $\left(4^{-1}\right)^{-6}$ and its equivalent expressions.

Q: What is the power rule in mathematics?

A: The power rule is a fundamental concept in mathematics that states that for any numbers $a$ and $b$ and any integers $m$ and $n$, the following equation holds:

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

This rule allows us to simplify expressions involving exponents.

Q: How do I simplify the expression $\left(4^{-1}\right)^{-6}$ using the power rule?

A: To simplify the expression $\left(4^{-1}\right)^{-6}$ using the power rule, we can multiply the exponents:

$$\left(4^{-1}\right)^{-6} = 4^{-1 \cdot -6} = 4^6$$

This simplification is based on the power rule, which states that the exponent of the inner expression is multiplied by the exponent of the outer expression.

Q: What is the rule for negative exponents?

A: The rule for negative exponents states that for any number $a$ and any integer $n$, the following equation holds:

$$a^{-n} = \frac{1}{a^n}$$

This rule allows us to rewrite negative exponents as fractions.

Q: How do I simplify the expression $\left(4^{-1}\right)^{-6}$ using the rule for negative exponents?

A: To simplify the expression $\left(4^{-1}\right)^{-6}$ using the rule for negative exponents, we can rewrite the negative exponent as a fraction:

$$\left(4^{-1}\right)^{-6} = \frac{1}{4^{-6}}$$

This simplification is based on the rule for negative exponents, which states that a negative exponent can be rewritten as a fraction.

Q: What are the equivalent expressions for $\left(4^{-1}\right)^{-6}$?

A: The equivalent expressions for $\left(4^{-1}\right)^{-6}$ are:

• $4^6$
• $\frac{1}{4^{-6}}$

These expressions are equivalent because they can be simplified to the same value using the power rule and the rule for negative exponents.

Q: How do I determine which expression is equivalent to $\left(4^{-1}\right)^{-6}$?

A: To determine which expression is equivalent to $\left(4^{-1}\right)^{-6}$, you can use the power rule and the rule for negative exponents to simplify each expression. If the simplified expression is the same as $\left(4^{-1}\right)^{-6}$, then it is an equivalent expression.

Q: What are some common mistakes to avoid when simplifying expressions involving exponents?

A: Some common mistakes to avoid when simplifying expressions involving exponents include:

• Not using the power rule to simplify expressions
• Not using the rule for negative exponents to rewrite negative exponents as fractions
• Not checking if the simplified expression is equivalent to the original expression

By avoiding these common mistakes, you can ensure that your simplifications are accurate and correct.

Conclusion

In conclusion, the expression $\left(4^{-1}\right)^{-6}$ can be simplified using the power rule and the rule for negative exponents. The equivalent expressions for $\left(4^{-1}\right)^{-6}$ are $4^6$ and $\frac{1}{4^{-6}}$. By following the rules of exponents and checking your simplifications, you can ensure that your expressions are accurate and correct.