Rewrite the fractions as powers with a negative exponent.

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for problems 6 and 7:

6. Simplify, and write as an index notation: $3 \times 3 \times 3 \times \frac{1}{3} \times 3 \times \frac{1}{3}$

Rewrite the fractions as powers with a negative exponent.
Recall that $\frac{1}{a} = a^{-1}$ . So, $\frac{1}{3} = 3^{-1}$ .
The expression becomes:
$3^1 \times 3^1 \times 3^1 \times 3^{-1} \times 3^1 \times 3^{-1}$
Apply the exponent rule $a^m \times a^n = a^{m+n}$ to combine the terms.
Add all the exponents:
$1 + 1 + 1 + (-1) + 1 + (-1) = 3 - 1 + 1 - 1 = 2$
Write the simplified expression in index notation.
$3^2$
The simplified expression in index notation is:
$\boxed{3^2}$
7. Evaluate: $\left(\frac{13^0}{5}\right)^2$
Evaluate the term $13^0$ .
Any non-zero number raised to the power of 0 is 1.
$13^0 = 1$
Substitute this value back into the expression.
$\left(\frac{1}{5}\right)^2$
Apply the exponent to both the numerator and the denominator.
$\frac{1^2}{5^2}$
Calculate the values.
$\frac{1 \times 1}{5 \times 5} = \frac{1}{25}$
The evaluated expression is:
$\boxed{\frac{1}{25}}$

Answer