Math Problem Statement

Evaluate the following expressions:

a. 10^810^7 b. 10^910^-5 c. 10^6/10^3 d. 10^8/10^-4

Solution

To evaluate the expressions involving powers of 10, we apply the laws of exponents. Specifically, for multiplication and division:

  1. Multiplication of powers of the same base:
    10a×10b=10a+b10^a \times 10^b = 10^{a+b}

  2. Division of powers of the same base:
    10a10b=10ab\frac{10^a}{10^b} = 10^{a-b}

Now, let's evaluate each expression:

a. 108×10710^8 \times 10^7
Using the multiplication rule:
108×107=108+7=101510^8 \times 10^7 = 10^{8+7} = 10^{15}

b. 109×10510^9 \times 10^{-5}
Using the multiplication rule:
109×105=109+(5)=10410^9 \times 10^{-5} = 10^{9 + (-5)} = 10^{4}

c. 106103\frac{10^6}{10^3}
Using the division rule:
106103=1063=103\frac{10^6}{10^3} = 10^{6-3} = 10^3

d. 108104\frac{10^8}{10^{-4}}
Using the division rule:
108104=108(4)=108+4=1012\frac{10^8}{10^{-4}} = 10^{8 - (-4)} = 10^{8+4} = 10^{12}

Final answers:

  • a. 101510^{15}
  • b. 10410^4
  • c. 10310^3
  • d. 101210^{12}

Do you want details or have any questions?

Here are 5 related questions to deepen your understanding:

  1. What is 105×10210^5 \times 10^{-2}?
  2. Simplify 1012106\frac{10^{12}}{10^6}.
  3. How would you simplify (103)4(10^3)^4?
  4. What is 107×103×10410^7 \times 10^3 \times 10^{-4}?
  5. Evaluate 109103\frac{10^9}{10^{-3}}.

Tip: When multiplying or dividing powers with the same base, always add or subtract the exponents!

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Math Problem Analysis

Mathematical Concepts

Exponents
Laws of Exponents
Powers of 10

Formulas

10^a * 10^b = 10^{a+b}
10^a / 10^b = 10^{a-b}

Theorems

Laws of Exponents

Suitable Grade Level

Grades 6-8