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Chapter 1: Problem 2

Manipulation of exponents a. \(106 \times 102=\)___________ b. \(\left(10^{2}\right)^{1 / 3}=\)___________ c. \(\left(10^{-5} \times 10^{3}\right): 10^{2}=\)___________

Short Answer

Expert verified
Question: Simplify the following expressions using the rules of exponents: a. \(106 \times 102\) b. \(\left(10^{2}\right)^{1 / 3}\) c. \(\left(10^{-5} \times 10^{3}\right): 10^{2}\) Answer: a. \(106 \times 102 = 100000\) b. \(\left(10^{2}\right)^{1 / 3} = 10^{2/3}\) c. \(\left(10^{-5} \times 10^{3}\right): 10^{2}= 10^{-4}\)

Step by step solution

01

Write numbers as powers of 10

First, we should write each number as a power of 10: \(106 = 10^2 \times 10^1\) and \(102= 10^2 \times 10^0\).
02

Apply the multiplication rule for exponents

Now, we will apply the multiplication rule for exponents, adding the exponents when the base is the same: \(106 \times 102 = (10^2 \times 10^1) \times (10^2 \times 10^0) = 10^{2+1} \times 10^{2+0} = 10^3 \times 10^2\).
03

Multiply the numbers

Now we can multiply: \(10^3 \times 10^2 = 10^{3+2} = 10^5 = 100000\). So, \(106 \times 102 = 100000\). #b. Exponent raised to another exponent#
04

Apply the power rule for exponents

The given expression is \(\left(10^{2}\right)^{1 / 3}\), and we need to apply the power rule for exponents here: \((a^m)^n = a^{m \cdot n}\). Therefore, \(\left(10^{2}\right)^{1 / 3} = 10^{2 \cdot (1 / 3)}\).
05

Calculate the resulting exponent

Next, we will calculate the exponent: \(2 \cdot (1 / 3) = 2 / 3\).
06

Write the simplified expression

Thus, we get the simplified expression: \(\left(10^{2}\right)^{1 / 3} = 10^{2/3}\). #c. Multiplying and dividing numbers with exponents#
07

Multiply the numbers

In the given expression, we first multiply the numbers: \(10^{-5} \times 10^{3}\). Apply the multiplication rule for exponents, adding the exponents when the base is the same: \(10^{-5} \times 10^{3} = 10^{-5+3} = 10^{-2}\).
08

Divide the result by the third number

Now, we will divide the result by \(10^2\): \(\frac{10^{-2}}{10^{2}}\).
09

Apply the division rule for exponents

Apply the exponent division rule, subtracting the exponent in the denominator from the exponent in the numerator: \(\frac{10^{-2}}{10^{2}} = 10^{-2-2} = 10^{-4}\).
10

Write the final simplified expression

Thus, the final simplified expression is: \(\left(10^{-5} \times 10^{3}\right): 10^{2}= 10^{-4}\).

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