Question

select all the expressions that are equivalent to $2^{-7} \cdot 2^{6}$.
$\frac{1}{2^{-1}}$ $\frac{1}{2^{-42}}$ $\frac{1}{2}$ $2^{-42}$

Explanation:

Step1: Simplify the original expression

Using the exponent rule \(a^m \cdot a^n = a^{m + n}\), for \(2^{-7} \cdot 2^{6}\), we have \(2^{-7 + 6}=2^{-1}\). And by the definition of negative exponents, \(2^{-1}=\frac{1}{2}\). Also, \(2^{-1}=\frac{1}{2^{1}}\), and \(\frac{1}{2^{-1}}=\frac{1}{\frac{1}{2}} = 2\)? Wait, no, wait. Wait, let's re - calculate. Wait, \(2^{-7}\cdot2^{6}=2^{-7 + 6}=2^{-1}=\frac{1}{2}\). Now let's check each option:

• Option 1: \(\frac{1}{2^{-1}}\). Since \(2^{-1}=\frac{1}{2}\), then \(\frac{1}{2^{-1}}=\frac{1}{\frac{1}{2}} = 2\), which is not equal to \(2^{-1}=\frac{1}{2}\). Wait, maybe I made a mistake. Wait, no, the original expression is \(2^{-7}\cdot2^{6}=2^{-1}=\frac{1}{2}\). Let's re - evaluate the options:

• Option \(\frac{1}{2^{-1}}\): \(2^{-1}=\frac{1}{2}\), so \(\frac{1}{2^{-1}} = 2\), which is not equal to \(\frac{1}{2}\). So this option is incorrect.

• Option \(\frac{1}{2^{-42}}\): \(2^{-42}=\frac{1}{2^{42}}\), so \(\frac{1}{2^{-42}}=2^{42}\), which is not equal to \(\frac{1}{2}\). So this option is incorrect.

• Option \(\frac{1}{2}\): This is equal to \(2^{-1}\), which is the result of \(2^{-7}\cdot2^{6}\) (since \(2^{-7}\cdot2^{6}=2^{-1}=\frac{1}{2}\)). So this option is correct.

• Option \(2^{-42}\): \(2^{-42}=\frac{1}{2^{42}}\), which is not equal to \(\frac{1}{2}\). So this option is incorrect.

Wait, maybe there was a mistake in the initial check. Let's start over.

Original expression: \(2^{-7}\cdot2^{6}\). Using the product rule for exponents \(a^m\times a^n=a^{m + n}\), so \(m=-7\), \(n = 6\), then \(2^{-7+6}=2^{-1}=\frac{1}{2}\).

Now check each option:

1. \(\frac{1}{2^{-1}}\): By the negative exponent rule, \(a^{-n}=\frac{1}{a^{n}}\), so \(2^{-1}=\frac{1}{2}\), then \(\frac{1}{2^{-1}}=\frac{1}{\frac{1}{2}}=2

eq\frac{1}{2}\). So this option is wrong.

1. \(\frac{1}{2^{-42}}\): \(2^{-42}=\frac{1}{2^{42}}\), so \(\frac{1}{2^{-42}} = 2^{42}

eq\frac{1}{2}\). Wrong.

1. \(\frac{1}{2}\): This is equal to \(2^{-1}\), which is the value of the original expression. Correct.

1. \(2^{-42}\): \(2^{-42}=\frac{1}{2^{42}}

eq\frac{1}{2}\). Wrong.

Wait, but the original image shows some checkmarks, maybe the user made a mistake in the checkmarks. But according to the correct calculation, only \(\frac{1}{2}\) is equivalent to \(2^{-7}\cdot2^{6}\).

Step2: Re - confirm the exponent rule

The product of two exponential terms with the same base \(a^m\times a^n=a^{m + n}\) is a fundamental exponent rule. For \(a = 2\), \(m=-7\), \(n = 6\), we get \(2^{-7+6}=2^{-1}=\frac{1}{2}\). The expression \(\frac{1}{2}\) is directly equal to \(2^{-1}\), so it is equivalent to the original expression.

Answer:

The expression equivalent to \(2^{-7}\cdot2^{6}\) is \(\frac{1}{2}\) (the third option in the given set of options).