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Chapter 5: Problem 85

Perform each indicated operation. Write each answer in scientific notation. $$ \frac{9.24 \times 10^{15}}{\left(2.2 \times 10^{-2}\right)\left(1.2 \times 10^{-5}\right)} $$

Short Answer

Expert verified
The answer is \(3.5 \times 10^{22}\).

Step by step solution

01

Solve the Inner Multiplication

First, solve the denominators of the expression by multiplying the two numbers: \( 2.2 \times 10^{-2} \) and \( 1.2 \times 10^{-5} \). Multiply the constants: \( 2.2 \times 1.2 = 2.64 \). Then add the exponents of 10: \( (-2) + (-5) = -7 \). Thus, the result of the denominator is \( 2.64 \times 10^{-7} \).
02

Set Up the Division

Replace the original denominator with the result from Step 1. Now, the expression is: \( \frac{9.24 \times 10^{15}}{2.64 \times 10^{-7}} \).
03

Divide the Constants

Divide the constants from the numerator and denominator: \( \frac{9.24}{2.64} \approx 3.5 \).
04

Subtract the Exponents

Subtract the exponent of the denominator from the exponent of the numerator: \(15 - (-7) = 15 + 7 = 22\).
05

Write the Final Answer in Scientific Notation

Combine the results from Step 3 and Step 4 to express the final answer in scientific notation: \( 3.5 \times 10^{22} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplication of Powers
When multiplying powers, one simple rule is applied: the exponents are added together. This occurs when the base remains constant. The base is the number that is raised to a power (the exponent). For example, in the expression \( a^m \cdot a^n \), the result would be \( a^{m+n} \).

Let's consider how this applies to scientific notation. As seen in the original exercise, when multiplying the scientific notations \( 2.2 \times 10^{-2} \) and \( 1.2 \times 10^{-5} \), you first multiply the coefficients: \( 2.2 \times 1.2 = 2.64 \).

Then, you add the exponents: \( -2 + (-5) = -7 \). Thus, the multiplication gives you \( 2.64 \times 10^{-7} \).
  • Multiply coefficients: simple arithmetic multiplication
  • Add exponents: \( a^m \cdot a^n = a^{m+n} \)
  • Result: combine new coefficient and power for the answer
Division of Powers
Dividing powers involves a straightforward rule similar to multiplication, but in this case, you subtract the exponents. This also applies when the bases are the same. For instance, in the expression \( \frac{a^m}{a^n} \), the result will be \( a^{m-n} \).

In the original exercise, once you have the product \( 2.64 \times 10^{-7} \) in the denominator, the expression rewrites into \( \frac{9.24 \times 10^{15}}{2.64 \times 10^{-7}} \).

Here, you first divide the numbers: \( \frac{9.24}{2.64} \approx 3.5 \).

Next, subtract the exponents: \( 15 - (-7) = 15 + 7 = 22 \). The result is \( 3.5 \times 10^{22} \).
  • Divide the coefficients: \( \frac{x}{y} \)
  • Subtract the exponents: \( a^m \div a^n = a^{m-n} \)
  • Compose the final scientific notation
Exponents Rules
Understanding how exponents work helps simplify and solve a vast range of mathematical problems. Exponent rules are a collection of the basic arithmetic steps used when performing operations involving powers of the same base.

Here鈥檚 a quick recap of the essential exponent rules you鈥檒l encounter:
  • **Product of Powers Rule**: When multiplying, add the exponents if base is same: \( a^m \cdot a^n = a^{m+n} \).

  • **Quotient of Powers Rule**: When dividing, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

  • **Power of a Power Rule**: Dealing with nested exponents, multiply them: \((a^m)^n = a^{m \cdot n}\).
  • **Zero Exponent Rule**: Any non-zero base raised to the zero power equals 1: \( a^0 = 1 \).

  • **Negative Exponent Rule**: The reciprocal rule for a negative exponent: \( a^{-n} = \frac{1}{a^n} \).
Exponents are crucial for the simplification of expressions and play a fundamental role in calculations involving scientific notation, algebra, and beyond.

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Most popular questions from this chapter

Factor. Assume that variables used as exponents represent positive integers. $$ x^{2 n}+10 x^{n}+16 $$

Factor. Assume that variables used as exponents represent positive integers. $$ 3 x^{2 n}-8 x^{n}+4 $$

Factor. Assume that variables used as exponents represent positive integers. $$ x^{2 n}-25 $$

Factor each polynomial completely. See Examples 1 through 12. $$ 2(x+4)^{2}+3(x+4)-5 $$

Factor. Assume that variables used as exponents represent positive integers. $$ 4 x^{2 n}-12 x^{n}+9 $$
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