Summary: Proportional Relationships

Table of Contents
1. Introduction
2. Power with Negative Exponents
3. Law of Exponents
4. Use of Exponents to Express Numbers in Standard Form
5. Standard form for large whole numbers
6. Standard form for small decimal numbers
7. Comparing Very Large and Very Small Numbers
8. Points to Remember
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Introduction

• Exponent shows how many times a number is multiplied by itself.
• Base is the number being multiplied.
• Power is the full expression of base with its exponent.

Power with Negative Exponents

• A negative exponent means the reciprocal: for non-zero a, a^-m = 1 / a^m.
• Negative exponents can be converted to positive by using the reciprocal.

Law of Exponents

• a^m × aⁿ = a^m+n
• a^m ÷ aⁿ = a^m-n
• (a^m)ⁿ = a^mn
• a^m × b^m = (ab)^m
• a⁰ = 1 (for a ≠ 0)
• a^m ÷ b^m = (a / b)^m

Use of Exponents to Express Numbers in Standard Form

1. Standard form for large whole numbers

• Step 1: Count digits to the right of the first non-zero digit.
• Step 2: Write the first digit (and following digits after a decimal point if there are more non-zero digits).
• Step 3: Multiply by 10 raised to the number counted in Step 1.

2. Standard form for small decimal numbers

• Step 1: Count digits from the decimal point to the last non-zero digit.
• Step 2: If one non-zero digit, write it and multiply by 10 to the negative of the count.
• Step 3: If two or more non-zero digits, write digits as a decimal starting after the first digit.
• Step 4: Subtract the number of digits after the decimal from the count in Step 1.
• Step 5: Multiply by 10 raised to the negative of the result from Step 4.

Comparing Very Large and Very Small Numbers

• Make exponents the same, then compare the coefficients to decide which number is larger or smaller.
• One method: divide coefficients and subtract exponents to get a single coefficient × 10^difference, then interpret the result.

Points to Remember

• Numbers with negative exponents follow the same laws of exponents listed above.
• Standard form (scientific form) expresses very large numbers with positive exponents and very small numbers with negative exponents.
• Coefficient is the number multiplied by a power of 10 when using standard form; compare coefficients after aligning exponents.