Pre-Algebra Workbook 11: Exponents & Scientific Notation

Expand the power as repeated multiplication: $3^4$ means $3 \cdot 3 \cdot 3 \cdot 3$

Write $3^4$ as repeated multiplication:

$$3^4 = 3 \cdot 3 \cdot 3 \cdot 3$$

Multiply step by step:

• $3 \cdot 3 = 9$
• $9 \cdot 3 = 27$
• $27 \cdot 3 = 81$

So $3^4 = 81$

The correct choice is C. $81$.

Expand $(-2)^3$ as $(-2) \cdot (-2) \cdot (-2)$ Remember: the product of two negatives is positive, and then multiply by another negative.

Write $(-2)^3$ as repeated multiplication:

$$(-2)^3 = (-2) \cdot (-2) \cdot (-2)$$

Multiply step by step:

• $(-2) \cdot (-2) = 4$ (negative times negative is positive)
• $4 \cdot (-2) = -8$

So $(-2)^3 = -8$

The correct choice is B. $-8$.

When multiplying powers with the same base, keep the base and add the exponents: $a^m \cdot a^n = a^{m+n}$

Use the product rule for exponents:

$$2^3 \cdot 2^4 = 2^{3+4} = 2^7$$

Evaluate $2^7$

$$2^7 = 128$$

So the simplified result is $128$

The correct choice is D. $128$.

For a quotient with the same base, subtract exponents: $\dfrac{a^m}{a^n} = a^{m-n}$

Apply the quotient rule:

$$\dfrac{x^5}{x^2} = x^{5-2} = x^3$$

So the simplified expression is $x^3$

The correct choice is A. $x^3$.

Use the power-of-a-power rule: $(a^m)^n = a^{m \cdot n}$

Apply the power-of-a-power rule:

$$(3^2)^3 = 3^{2 \cdot 3} = 3^6$$

Evaluate $3^6$

$$3^6 = 3^3 \cdot 3^3 = 27 \cdot 27 = 729$$

So $(3^2)^3 = 729$

The correct choice is C. $729$.

The bases are different, so the exponent rules for combining exponents do not apply across $2$ and $5$ Evaluate each power separately, then multiply the results.

Evaluate each power first:

• $2^3 = 8$
• $5^2 = 25$

Now multiply:

$$8 \cdot 25 = 200$$

So the simplified value is $200$

The correct choice is B. $200$.

Move the decimal so you have a number between $1$ and $10$ then count how many places you moved it. That count becomes the positive exponent of $10$

Start with $4{,}500{,}000$ and place the decimal after the first nonzero digit:

$$4{,}500{,}000 = 4.5 \times 1{,}000{,}000$$

We moved the decimal $6$ places to the left:

$$1{,}000{,}000 = 10^6$$

So in scientific notation:

$$4{,}500{,}000 = 4.5 \times 10^6$$

The correct choice is A. $4.5 \times 10^6$.

Move the decimal to get a number between $1$ and $10$ Count how many places you move it to the right; that count becomes a negative exponent: $a \times 10^{-n}$

Place the decimal so the number is between $1$ and $10$

$$0.00032 = 3.2 \times 0.0001$$

We moved the decimal $4$ places to the right, so the exponent is $-4$

$$0.0001 = 10^{-4}$$

So in scientific notation:

$$0.00032 = 3.2 \times 10^{-4}$$

The correct choice is C. $3.2 \times 10^{-4}$.

Multiply the numbers in front and use the product rule on the powers of $10$ $(3 \times 2) \times 10^{4+3}$

Multiply the coefficients and then the powers of $10$

$$(3 \times 10^4)(2 \times 10^3) = (3 \cdot 2) \times 10^{4+3}$$

Compute each part:

$$3 \cdot 2 = 6, \quad 4 + 3 = 7$$

so

$$(3 \times 10^4)(2 \times 10^3) = 6 \times 10^7$$

This is already in scientific notation.

The correct choice is B. $6 \times 10^7$.

Divide the coefficients and subtract the exponents: $\dfrac{4.8}{6} \times 10^{5-2}$ Then adjust the coefficient if needed so it is between $1$ and $10$

Separate the fraction into coefficients and powers of $10$

$$\dfrac{4.8 \times 10^5}{6 \times 10^2} = \left(\dfrac{4.8}{6}\right) \times 10^{5-2}$$

Compute the coefficient and exponent:

$$\dfrac{4.8}{6} = 0.8, \quad 5 - 2 = 3$$

So we have $0.8 \times 10^3$ To write this in scientific notation, move the decimal one place to make $8.0$

$$0.8 \times 10^3 = 8.0 \times 10^2$$

So in scientific notation, the result is $8 \times 10^2$

The correct choice is C. $8 \times 10^2$.