Question

part three: review! read each question carefully and be sure to show all of your work.

1. simplify the following exponential expressions.

a. (9⁵)(9³) = 9⁸
b. 2¹ / 2³
c. (5²)⁵ = 5¹⁰

1. which value in scientific notation is the best estimate for the following value: 8,78,555,321

a. 8 × 10⁸
b. 9 × 10⁸
c. 9 × 10⁻⁸
d. 8 × 10⁻⁸

1. answer each word problem below. remember that when you see the word about you can round values to make them easier to multiply/divide.

• before you do any math, make sure you determine if you are supposed to multiply or divide.

Explanation:

Problem 1a: Simplify \((9^5)(9^3)\)

Step 1: Recall the exponent rule for multiplication

When multiplying two powers with the same base, we add the exponents: \(a^m \cdot a^n = a^{m + n}\). Here, the base \(a = 9\), \(m = 5\), and \(n = 3\).
\((9^5)(9^3)=9^{5 + 3}\)

Step 2: Add the exponents

\(5+3 = 8\), so \(9^{5 + 3}=9^8\)

Step 1: Recall the exponent rule for division

When dividing two powers with the same base, we subtract the exponents: \(\frac{a^m}{a^n}=a^{m - n}\). Here, the base \(a = 2\), \(m = 1\), and \(n = 3\).
\(\frac{2^1}{2^3}=2^{1 - 3}\)

Step 2: Subtract the exponents

\(1-3=-2\), so \(2^{1 - 3}=2^{-2}\). We can also write this as \(\frac{1}{2^2}=\frac{1}{4}\) (using the rule \(a^{-n}=\frac{1}{a^n}\))

Step 1: Recall the exponent rule for a power of a power

When raising a power to a power, we multiply the exponents: \((a^m)^n=a^{m\times n}\). Here, the base \(a = 5\), \(m = 2\), and \(n = 5\).
\((5^2)^5 = 5^{2\times5}\)

Step 2: Multiply the exponents

\(2\times5 = 10\), so \(5^{2\times5}=5^{10}\)

Answer:

\(9^8\)

Problem 1b: Simplify \(\frac{2^1}{2^3}\)