Powers, Indices & Standard Form

How do you simplify a multiplication expression with the same base, like aᵐ × aⁿ?

aᵐ⁺ⁿ (Add the powers). Example: 2⁴ × 2³ = 2⁷.

How do you simplify a division expression with the same base, like aᵐ ÷ aⁿ?

aᵐ⁻ⁿ (Subtract the powers). Example: 5⁸ ÷ 5² = 5⁶.

How do you simplify an expression raised to another power, like (aᵐ)ⁿ?

aᵐⁿ (Multiply the powers). Example: (3²)⁴ = 3⁸.

What is the result when any non-zero number is raised to the power of 0 (a⁰)?

1 (Any non-zero number raised to power 0 equals 1).

What are the first 15 square numbers (from 1² up to 15²)?

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.

What are the values of 1³, 2³, 3³, 4³, 5³, and 10³?

1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 10³ = 1000.

What are the numerical values of 10⁰, 10¹, 10², 10³, 10⁶, and 10⁹?

10⁰ = 1, 10¹ = 10, 10² = 100, 10³ = 1,000, 10⁶ = 1,000,000, 10⁹ = 1,000,000,000.

In the expression 10ⁿ, what does the exponent n represent visually in the expanded number?

It indicates the total number of zeros after the digit 1.

What is the standard mathematical structure for writing a number in Standard Form?

A × 10ⁿ (where 1 ≤ A < 10 and n is an integer).

How do you express 54,000,000 in Standard Form?

5.4 × 10⁷ (Decimal moves left 7 places, positive power).

How do you express 0.000062 in Standard Form?

6.2 × 10⁻⁵ (Decimal moves right 5 places, negative power).

What procedure do you follow to multiply two numbers written in standard form, like (A × 10ᵃ) × (B × 10ᵇ)?

Multiply front numbers (A × B) and ADD powers (a + b). Adjust to standard form if front number ≥ 10.

What procedure do you follow to divide two numbers written in standard form, like (A × 10ᵃ) ÷ (B × 10ᵇ)?

Divide front numbers (A ÷ B) and SUBTRACT powers (a - b). Adjust to standard form if front number < 1.

How is a negative power of ten (10⁻ⁿ) written in fractional form?

1 ÷ (10ⁿ)

What are the decimal equivalents for 10⁻¹, 10⁻², and 10⁻³?

10⁻¹ = 0.1, 10⁻² = 0.01, 10⁻³ = 0.001.

What equivalent operation can you perform instead of multiplying a number by 10⁻ⁿ?

Multiplying by 10⁻ⁿ is identical to DIVIDING by 10ⁿ. Example: 450 × 10⁻² = 450 ÷ 100 = 4.5.

What equivalent operation can you perform instead of dividing a number by 10⁻ⁿ?

Dividing by 10⁻ⁿ is identical to MULTIPLYING by 10ⁿ. Example: 7 ÷ 10⁻³ = 7 × 1000 = 7000.