What is the definition of an exponent (or power)?

An exponent (or power) indicates how many times a number, called the base, is multiplied by itself. For example, in 3², 3 is the base and 2 is the exponent, meaning 3 * 3 = 9.

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

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

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

When dividing powers with the same base, you subtract the exponents. For example, b⁵/ b² = b^{(5 - 2)} = b³.

Solve for x: 5^x = 125.

Hint: Express 125 as a power of 5.

Since 125 can be expressed as 5³, we have 5^x = 5³. Therefore, x = 3.

What is the rule for raising a power to a negative exponent?

A power with a negative exponent can be expressed as the reciprocal of the base raised to the opposite positive exponent. For example, a^-2 = 1/a².

Simplify: x² * x⁴.

Hint: Use the multiplication rule for exponents.

Using the multiplication rule, we add the exponents: x² * x⁴ = x⁽²⁺⁴⁾= x⁶.

Evaluate: (3² * 3³) / 3⁴.

Hint: Apply the division rule for exponents.

Using the division rule, we subtract the exponents: (3²* 3³) / 3⁴ = 3^{(2 + 3 - 4)} = 3¹ = 3.

What is the value of 10⁰?

Hint: Recall the rule for any non-zero number raised to the power of zero.

According to the rule, any non-zero number raised to the power of zero equals 1. Thus, 10⁰ = 1.

What is the result of (x⁵)⁴?

Hint: Use the power of a power rule.

Using the power of a power rule, (x⁵)⁴ = x^(5*4) = x²⁰.

What is 6⁻¹?

Hint: Recall the rule for negative exponents.

Using the negative exponent rule, 6⁻¹ = 1/6.