Question

select all the expressions that are equivalent to $7^2 \cdot 6^2$.
$42^4$ $\frac{42^9}{42^7}$
$42^2$ $\frac{1}{42^2}$

Explanation:

Step1: Simplify \(7^2 \cdot 6^2\)

Using the property \(a^n \cdot b^n=(ab)^n\), we have \(7^2 \cdot 6^2=(7\times6)^2 = 42^2\). Also, \((ab)^n=a^n b^n\), and if we consider exponents, \(7^2\cdot6^2 = 7^2\times6^2=(7\times6)^2 = 42^2\), and also, \(7^2\cdot6^2=(7\times6)^2 = 42^2\), and \(42^2\times42^2=42^{2 + 2}=42^4\) (if we square \(42^2\)), and using the quotient rule of exponents \(\frac{a^m}{a^n}=a^{m - n}\), \(\frac{42^9}{42^7}=42^{9-7}=42^2\), which is equal to \(7^2\cdot6^2\). Wait, first, let's re - evaluate:

First, \(7^2\cdot6^2=(7\times6)^2=42^2\) (by the power of a product property \((ab)^n=a^n b^n\)).

Now let's check each option:

1. \(42^4\): \(42^2\times42^2 = 42^{2 + 2}=42^4\), and \(7^2\cdot6^2 = 42^2\), so \(42^2\times42^2=(7^2\cdot6^2)\times(7^2\cdot6^2)=(7^2\cdot6^2)^2\), which is \(42^4\). So \(42^4\) is equivalent to \((7^2\cdot6^2)^2\), but wait, the original expression is \(7^2\cdot6^2 = 42^2\). Wait, maybe I made a mistake. Wait, \(7^2\cdot6^2=(7\times6)^2 = 42^2\). Now:

- Option 1: \(42^4\): \(42^4=(42^2)^2=(7^2\cdot6^2)^2\), which is not equal to \(7^2\cdot6^2\) unless we square the original expression. Wait, no, maybe the problem is that \(7^2\cdot6^2 = 42^2\), and \(42^2\times42^2=42^4\), but the original expression is \(42^2\). Wait, maybe the initial step was wrong. Wait, \(7^2\cdot6^2=(7\times6)^2 = 42^2\), and also, \(7^2\cdot6^2 = 7^2\times6^2\), and if we consider \( (7^2\cdot6^2)^2=7^4\cdot6^4 = 42^4\), but the original expression is \(7^2\cdot6^2\). Wait, maybe the user's check marks are wrong, but let's re - do:

Wait, the original expression is \(7^2\cdot6^2\). Let's compute its value: \(7^2 = 49\), \(6^2=36\), \(49\times36 = 1764\).

Now check each option:

- \(42^4\): \(42^4=42\times42\times42\times42 = 1764\times1764=3111696\). \(1764 eq3111696\), so \(42^4\) is not equivalent. Wait, I see my mistake earlier. The property \(a^n\cdot b^n=(ab)^n\) gives \(7^2\cdot6^2=(7\times6)^2 = 42^2\), not \(42^4\). So my first step was wrong. Let's correct:

Step1: Correct simplification of \(7^2\cdot6^2\)

Using the power of a product rule: \(a^n\times b^n=(a\times b)^n\). So \(7^2\times6^2=(7\times6)^2 = 42^2\).

Step2: Evaluate each option

- Option 1: \(42^4\): \(42^4=(42^2)^2\). Since \(7^2\cdot6^2 = 42^2\), then \((7^2\cdot6^2)^2=42^4\). But the original expression is \(7^2\cdot6^2\), not its square. Wait, but maybe the question is considering that \(7^2\cdot6^2\) can be part of a larger exponentiation. Wait, no, let's calculate the numerical values:

\(7^2\cdot6^2=49\times36 = 1764\)

\(42^2 = 1764\), \(42^4=42\times42\times42\times42 = 1764\times1764 = 3111696\), \(\frac{42^9}{42^7}=42^{9 - 7}=42^2=1764\), \(\frac{1}{42^2}=\frac{1}{1764}\)

So:

- \(42^2\): Equals \(7^2\cdot6^2\) (correct)
- \(\frac{42^9}{42^7}\): Using \(\frac{a^m}{a^n}=a^{m - n}\), \(42^{9-7}=42^2\) (correct)
- \(42^4\): \(42^4=(42^2)^2 = 1764^2 eq1764\) (incorrect)
- \(\frac{1}{42^2}\): \(\frac{1}{1764} eq1764\) (incorrect)

Wait, the original check marks in the image are wrong. Let's do it properly:

1. \(7^2\cdot6^2=(7\times6)^2 = 42^2\)
2. Check \(42^4\): \(42^4=(42^2)^2\), and \(7^2\cdot6^2 = 42^2\), so \(42^4=(7^2\cdot6^2)^2 eq7^2\cdot6^2\)
3. Check \(\frac{42^9}{42^7}\): By the quotient rule \(\frac{a^m}{a^n}=a^{m - n}\), so \(\frac{42^9}{42^7}=42^{9 - 7}=42^2\), which is equal to \(7^2\cdot6^2\) (correct)
4. Check \(42^2\): Equal to \(7^2\cdot6^2\) (correct)
5. Check \(\frac{1}{42^2}\): Reciprocal of \(42^2\), not equal to \(42^2\) (incorrect)

So the correct expressions are \(42^2\) and \(\frac{42^9}{42^7}\) and \(42^2\) (wait, \(42^2\) is the original simplified form, \(\frac{42^9}{42^7}=42^2\)).

Wait, let's re - express the original problem: We need to find all expressions equivalent to \(7^2\cdot6^2\).

First, \(7^2\cdot6^2=(7\times6)^2 = 42^2\) (power of a product property: \((ab)^n=a^n b^n\))

Now:

- \(42^4\): \(42^4=(42^2)^2=(7^2\cdot6^2)^2\). This is the square of the original expression, so it's not equivalent unless we square the original, so it's incorrect.
- \(\frac{42^9}{42^7}\): Using the quotient rule for exponents \(\frac{a^m}{a^n}=a^{m - n}\), we get \(42^{9 - 7}=42^2\), which is equal to \(7^2\cdot6^2\). So this is correct.
- \(42^2\): This is equal to \(7^2\cdot6^2\) (by power of a product), so this is correct.
- \(\frac{1}{42^2}\): This is the reciprocal of \(42^2\), so it's not equal to \(42^2\), so it's incorrect.

So the correct options are \(42^2\) and \(\frac{42^9}{42^7}\) and \(42^2\) (the \(42^2\) option). Wait, in the image, the check marks were wrong, but let's answer based on correct mathematical principles.

Answer:

The correct expressions are \(42^2\) and \(\frac{42^9}{42^7}\) (and \(42^2\) as a separate option, but in the options given, the correct ones are \(42^2\) (the third option) and \(\frac{42^9}{42^7}\) (the second option), and also, if we consider that \(7^2\cdot6^2 = 42^2\), and \(42^2\times42^2=42^4\) is not equal to \(7^2\cdot6^2\), \(\frac{1}{42^2}\) is not equal. So the correct options are:

• \(42^2\) (the third option in the image, the one with \(42^2\))
• \(\frac{42^9}{42^7}\) (the second option in the image)