What is an exponent?

An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in 2³, 2 is the base and 3 is the exponent, meaning 2 * 2 * 2 = 8.

What is the rule for multiplying numbers with the same base?

When multiplying numbers with the same base, you add the exponents. For example, a² * a³ = a^{(2 + 3)} = a⁵.

How do you simplify the expression: (x²)³?

To simplify (x²)³, you multiply the exponents. So, (x²)³ = x^(2*3) = x⁶.

Solve for x: 3^x = 81.

Hint: Rewrite 81 as a power of 3.

Since 81 = 3⁴, we can rewrite the equation as 3^x = 3⁴. Therefore, x = 4.

What is the value of 5⁰?

Any non-zero number raised to the power of zero is 1, so 5⁰ = 1.

What is the rule for dividing numbers with the same base?

When dividing numbers with the same base, you subtract the exponents. For example, a⁵ / a² = a^{(5 - 2)} = a³.

Simplify the expression: (xy)².

To simplify (xy)², apply the power to both the x and y: (xy)² = x²y².

If a = 2 and b = 3, what is the value of a²b³?

Substituting the values, a²b³ = 2² * 3³ = 4 * 27 = 108.

Evaluate: (2⁴)².

Hint: Use the power of a power rule.

Using the power of a power rule, (2⁴)² = 2^(4*2) = 2⁸ = 256.

What is the result of 10² * 10⁻³?

Using the multiplication rule, 10² * 10⁻³ = 10^{(2 + (-3))} = 10⁻¹ = 1/10.