What is an exponent?

An exponent indicates how many times a base is multiplied by itself. For example, in 3², the exponent is 2, meaning 3 × 3.

What is the formula for multiplying powers with the same base?

To multiply powers with the same base, add the exponents. For example, aⁿ × aᵐ = aⁿ⁺ᵐ.

How do you simplify a negative exponent?

A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, x⁻² = 1/x².

What is the formula for dividing powers with the same base?

To divide powers with the same base, subtract the exponents. For example, aⁿ / aᵐ = aⁿ⁻ᵐ.

What is an important rule for zero exponents?

Any base raised to the power of zero equals one, as long as the base is not zero. For example, a⁰ = 1.

What is the formula for raising a power to another power?

When raising a power to another power, multiply the exponents. For example, (aⁿ)ᵐ = aⁿᵐ.

How do you simplify the expression (3²)³?

Using the power of a power rule, (3²)³ = 3²×³ = 3⁶ = 729.

Evaluate: 2⁴ × 2².

Using the multiplication rule, 2⁴ × 2² = 2⁴⁺² = 2⁶ = 64.

What is the formula for powers of a product?

When raising a product to a power, distribute the exponent to each factor. For example, (ab)ⁿ = aⁿbⁿ.

Simplify: (5/2)².

Using the power of a fraction rule, (5/2)² = 5² / 2² = 25/4.

Solve for x in the equation 2⁴ = 16.

Since 2⁴ equals 16, x = 4.

What is the relationship between fractional exponents and roots?

A fractional exponent represents a root. For example, a^(1/n) equals the nth root of a.

Evaluate: 27^(2/3).

Using the fractional exponent rule, 27^(2/3) = (3³)^(2/3) = 3^(3×(2/3)) = 3² = 9.

Simplify: 4x²y⁴ / 2xy².

Using the division rule, 4x²y⁴ / 2xy² = (4/2)(x²/x)(y⁴/y²) = 2x¹y² = 2xy².

How do you express 64^(1/6) in radical form?

64^(1/6) can be expressed as the sixth root of 64, which is √[6]{64} = 2.

When simplifying √(x²y), what is the result?

The result is x√y, as √(x²) = x.

If a = 3 and b = 2, calculate a²b³.

Substituting values, a²b³ = 3² × 2³ = 9 × 8 = 72.