Question

select all the expressions that are equivalent to $4^{-5} \cdot 10^{-5}$.
$40^{-10}$
$\frac{40^{-8}}{40^{-3}}$
$40^{25}$
$40^{-4} \cdot 40^{-1}$

Explanation:

Step1: Simplify the original expression

First, recall the exponent rule \(a^m \cdot b^m=(ab)^m\). For \(4^{-5}\cdot10^{-5}\), we can apply this rule: \(4^{-5}\cdot10^{-5}=(4\times10)^{-5}=40^{-5}\)? Wait, no, wait. Wait, the rule is \(a^n\cdot b^n=(ab)^n\), but here the exponents are the same (-5), so \(4^{-5}\cdot10^{-5}=(4\times10)^{-5}=40^{-5}\)? Wait, but let's check the options. Wait, maybe I made a mistake. Wait, no, let's re - examine.

Wait, the original expression is \(4^{-5}\cdot10^{-5}\). Let's first simplify the original expression:

Using the property \(a^m\times b^m=(a\times b)^m\), so \(4^{-5}\times10^{-5}=(4\times10)^{-5}=40^{-5}\).

Now let's check each option:

Option 1: \(40^{-10}\)

Is \(40^{-5}=40^{-10}\)? No, because if \(40^x = 40^y\), then \(x = y\) (for \(a>0,a
eq1\)). So \(40^{-5}
eq40^{-10}\). So this option is incorrect.

Option 2: \(\frac{40^{-8}}{40^{-3}}\)

Using the exponent rule \(\frac{a^m}{a^n}=a^{m - n}\), so \(\frac{40^{-8}}{40^{-3}}=40^{-8-(-3)}=40^{-8 + 3}=40^{-5}\). So this option is correct.

Option 3: \(40^{25}\)

\(40^{25}\) is a positive exponent with a large power, while our original expression is \(40^{-5}\), so \(40^{25}
eq40^{-5}\). This option is incorrect.

Option 4: \(40^{-4}\cdot40^{-1}\)

Using the property \(a^m\times a^n=a^{m + n}\), \(40^{-4}\times40^{-1}=40^{-4-1}=40^{-5}\). So this option is correct.

Wait, but the original check - marks in the image are wrong. Let's correct the analysis:

1. Analyze \(4^{-5}\cdot10^{-5}\):

Using the exponent rule \(a^n\cdot b^n=(ab)^n\), so \(4^{-5}\cdot10^{-5}=(4\times10)^{-5}=40^{-5}\).

1. Analyze \(40^{-10}\):

Since \(40^{-5}
eq40^{-10}\) (because the exponents are different), this expression is not equivalent.

1. Analyze \(\frac{40^{-8}}{40^{-3}}\):

Using \(\frac{a^m}{a^n}=a^{m - n}\), we have \(\frac{40^{-8}}{40^{-3}}=40^{-8-(-3)}=40^{-5}\), which is equivalent to \(4^{-5}\cdot10^{-5}\).

1. Analyze \(40^{25}\):

\(40^{25}\) is a positive - exponent term with a large magnitude, while \(4^{-5}\cdot10^{-5}=40^{-5}\) is a small positive fraction (since the exponent is negative). So they are not equivalent.

1. Analyze \(40^{-4}\cdot40^{-1}\):

Using \(a^m\cdot a^n=a^{m + n}\), \(40^{-4}\cdot40^{-1}=40^{-4 + (-1)}=40^{-5}\), which is equivalent to \(4^{-5}\cdot10^{-5}\).

So the correct expressions are \(\frac{40^{-8}}{40^{-3}}\) and \(40^{-4}\cdot40^{-1}\).

Answer:

The equivalent expressions are \(\boldsymbol{\frac{40^{-8}}{40^{-3}}}\) and \(\boldsymbol{40^{-4}\cdot40^{-1}}\)