1. The problem asks us to describe what raising a number to a power means and explain the difference between $-7^2$ and $(-7)^2$. 2. Raising a number to a power means multiplying that number by itself a certain number of times. The exponent tells how many times the base is multiplied. For example, $3^4$ means $3 \times 3 \times 3 \times 3$. 3. Now, let's look at the difference between $-7^2$ and $(-7)^2$: - $-7^2$ means the negative of $7^2$. Since $7^2 = 49$, $-7^2 = -49$. - $(-7)^2$ means $-7$ multiplied by itself: $(-7) \times (-7) = 49$. 4. The parentheses change the base being squared. Without parentheses, only 7 is squared, then negated. With parentheses, the negative sign is included in the base and the squared result is positive. 5. The next problem is to solve for $x$ in $16^{3x-2} = (1/4)^{5-x}$ using laws of indices. 6. First rewrite bases as powers of 2: - $16 = 2^4$ so $16^{3x-2} = (2^4)^{3x-2} = 2^{4(3x-2)} = 2^{12x-8}$. - $\frac{1}{4} = 4^{-1}$ and $4 = 2^2$, so $\frac{1}{4} = (2^2)^{-1} = 2^{-2}$. Therefore, $(1/4)^{5-x} = (2^{-2})^{5-x} = 2^{-2(5-x)} = 2^{-10+2x}$. 7. Now the equation is: $$2^{12x-8} = 2^{-10 + 2x}$$ 8. Since bases are equal, set exponents equal: $$12x - 8 = -10 + 2x$$ 9. Solve for $x$: $12x - 8 = -10 + 2x$ $12x - 2x = -10 + 8$ $10x = -2$ $$x = \frac{-2}{10} = -0.2$$ 10. So the solution to the equation is $x = -0.2$.