Exponent Laws Practice

These questions follow Exponent Laws and Rational Exponents. Everything here yields to the laws plus the two meanings you built — negative means reciprocal, fractional means root. Exact values only; the calculator sits this one out until the final question.

Numbers first

1. Evaluate $2^{-3}$ and $5^0\cdot 3^{-2}$.
2. Evaluate $4^{3/2}$ and $27^{-1/3}$.
3. Evaluate ${\left(\frac{16}{25}\right)}^{1/2}$ and $8^{2/3}$.

Answer 1

$2^{-3}=\frac{1}{2^3}=\frac{1}{8}$ — reciprocal, not negative. $5^0\cdot 3^{-2}=1\cdot \frac{1}{9}=\frac{1}{9}$. Anything nonzero to the power 0 is 1; the descending pattern demands it.

Answer 2

Root first, power second: $4^{3/2}={\left(\sqrt{4}\right)}^3=2^3=8$. And $27^{-1/3}=\frac{1}{27^{1/3}}=\frac{1}{3}$ — unpack in order: reciprocal, then root.

Answer 3

${\left(\frac{16}{25}\right)}^{1/2}=\frac{\sqrt{16}}{\sqrt{25}}=\frac{4}{5}$. $8^{2/3}={\left(\sqrt[3]{8}\right)}^2=2^2=4$. Taking the root first kept every number single-digit.

Now with variables

4. Simplify $x^5\cdot x^{-2}$ and $\frac{{\left(y^3\right)}^4}{y^7}$.
5. Simplify ${\left(x^6{y}^3\right)}^{1/3}$.
6. Simplify $x^3÷x^{1/2}$.

Answer 4

$x^5\cdot x^{-2}=x^{5+(-2)}=x^3$. $\frac{(y^3{)}^4}{y^7}=\frac{y^{12}}{y^7}=y^5$. Add across products, multiply across powers of powers, subtract across quotients — each rebuildable by counting factors.

Answer 5

The exponent $\frac{1}{3}$ distributes across the product: $x^{6/3}{y}^{3/3}=x^{2}y$. A quick check with $x=y=1$ won’t catch much — try $x=2,y=1$: ${\left(64\right)}^{1/3}=4=2^2$. ✓

Answer 6

Subtract exponents: $x^{3-1/2}=x^{5/2}$. Fractional exponents obey the same quotient law as integers — that was the whole point of defining them the way we did.

Rewriting and reality

7. Rewrite $9^x$ as a power of 3, and $8^x$ as a power of 2.
8. Now the calculator: evaluate $1.01^{120}$, and write one sentence about why a number so close to 1 produced a result so far from 1.

Answer 7

$9^x={\left(3^2\right)}^x=3^{2x}$ and $8^x={\left(2^3\right)}^x=2^{3x}$ — power of a power. Same functions, new bases: the costume change that lets exponential models be compared on equal footing.

Answer 8

$1.01^{120}\approx 3.30$. Multiplying by 1.01 gains only 1% — but doing it 120 times compounds each gain on top of the last, and the total more than triples. Hold that thought: in Money Over Time, “1% per period for 120 periods” is called a loan.