Challenge Yourself: Indices Test

Any non-zero number raised to the power of 0 is always 1. For example, x^0 = x^(n-n) = x^(n)/x^(n) = 1. This rule works for all numbers except 0, as 0^0 is undefined.

When multiplying terms with the same base, you add their powers. Here, the base is b, and the powers are 3 and 4. So, b^3 x b^4 = b^(3+4) = b^7. This rule works for all exponents as long as the base remains the same.

When raising a power to another power, you multiply the exponents. Here, (a^4)^5 becomes a^(4x5) = a^20. Since a^20 is not equal to a^9, the statement is false. Always remember to multiply exponents when a power is raised to another power.

When raising a power to another power, multiply the exponents. Here, (8^-2)3 becomes 8^((-2)x3) = 8^-6. This follows the rule of multiplying exponents when a power is raised to another power.

Add the exponents for the base c. Here, (2n+3) + (3n-1) = 5n+2. So, c^2n+3 x c^3n-1 = c5n+2. The base c does not change, and only the powers are added together.

Multiply terms with the same base by adding their powers. Here, the powers are n and 2n-1. Add them: n + (2n-1) = 3n-1. So, 4^n x 4^2n-1 = 4^3n-1. The base remains 4 throughout.

First, combine r^3 and r^2 by adding their powers since the base is the same. This gives r^(3+2) = r^5. Multiply by 2, which remains a coefficient. The final answer is 2r^5. The 2 does not affect the exponent of r.

Combine the powers of a by adding the exponents: 2 + 3 - 1 = 4. This gives a^4. Multiply the constants 5 and 2 to get 10. The final answer is 10a^4. The coefficients and exponents are handled separately.

Apply the -2 power to both terms. For p^3, multiply 3 x -2 to get p^-6. For q^-5, multiply -5 x -2 to get q^10. The result is p^-6q^10. Negative exponents mean reciprocal values if simplified further.

Raising a negative number to an even power always results in a positive value. Here, (-5)^4 = (-5) x (-5) x (-5) x (-5) = 625. Negative signs cancel out in pairs. If the power was odd, the result would be negative.