Power Of A Power

Simplify (a^b)^c using the power of a power exponent rule. Enter base and two exponents to apply (a^b)^c = a^(bxc) with step-by-step results and formula breakdown. Free online tool.

Simplify power of power

Base (a):

First Exponent (b):

Second Exponent (c):

About This Calculator

The Power of a Power Calculator simplifies expressions of the form (a^b)^c using the fundamental exponent rule that states (a^b)^c = a^bxc. This calculator is designed for students learning exponent rules in algebra, teachers preparing lesson materials, and professionals who need quick verification of exponent calculations. Simply enter the base value and two exponents, and the calculator instantly shows the simplified expression a^(bxc), the final numeric result, and the complete expression in standard form.

The power of a power rule is one of the core exponent laws, alongside the product of powers rule (a^b x a^c = a^b+c) and the quotient of powers rule (a^b ÷ a^c = a^b-c). Understanding these rules is essential for simplifying algebraic expressions, working with polynomial functions, and solving exponential equations. The rule works for all real number exponents -- positive, negative, fractional, and zero -- making it a versatile tool for everything from basic arithmetic to advanced calculus.

Key properties of the power of a power rule: Keep the base unchanged; multiply the two exponents together; the result is the base raised to the product exponent. For example, (2³)⁴ = 2^3x4 = 2¹² = 4,096. This can also be verified by computing the inner power first: 2³ = 8, then 8⁴ = 8 x 8 x 8 x 8 = 4,096. Both methods yield the same result, confirming the rule's validity.

This calculator supports step-by-step learning by clearly showing the intermediate simplified form before giving the final computed value. The shareable URL feature allows you to save or send specific calculations, making it useful for classroom demonstrations and homework collaboration.

Frequently Asked Questions

What is the power of a power rule in exponents?

The power of a power rule states that when you raise one exponent to another exponent, you multiply the exponents together while keeping the base unchanged. Mathematically, (a^b)^c = a^(bxc). For example, (2^3)^4 = 2^(3x4) = 2^12 = 4,096.

Does the power of a power rule work with negative exponents?

Yes, the power of a power rule applies to negative exponents as well. Simply multiply the exponents together as usual. For example, (2^-3)^2 = 2^(-3x2) = 2^-6 = 1/64. The rule holds for any real number exponent, whether positive, negative, or fractional.

Does the power of a power rule work with fractional exponents?

Yes, the rule works with fractional exponents too. When you raise a power to a fractional exponent, multiply the exponents as fractions. For example, (4^2)^(1/2) = 4^(2x1/2) = 4^1 = 4. This is equivalent to taking the square root of 4^2.

What is the difference between power of a power and product of powers?

The power of a power rule (a^b)^c = a^(bxc) applies when an exponent is raised to another exponent. The product of powers rule a^b x a^c = a^(b+c) applies when multiplying two powers with the same base. They are different operations: one multiplies exponents, the other adds them.

How do you calculate (3^2)^3 by hand?

First apply the power of a power rule: (3^2)^3 = 3^(2x3) = 3^6. Then compute 3^6 = 3 x 3 x 3 x 3 x 3 x 3 = 729. Alternatively, compute the inner power first: 3^2 = 9, then raise to the outer exponent: 9^3 = 9 x 9 x 9 = 729.

Is this power of a power calculator free to use?

Yes, all calculators on Calculy are completely free to use with no registration required. You can compute unlimited power of a power expressions, share results via URL, and use the tool for homework, exam preparation, or teaching.

What is (a^b)^c equal to in simplified form?

Using the power of a power rule, (a^b)^c simplifies to a^(bxc). This means you keep the same base a, and the new exponent is the product of b and c. For example, (5^2)^4 = 5^(2x4) = 5^8. The calculator shows both the simplified expression a^(bxc) and the final numeric result.

Can the power of a power calculator handle zero exponents?

Yes, any real number exponent is supported. If either exponent is 0, the result follows standard exponent rules: a^0 = 1. For example, (2^3)^0 = 2^(3x0) = 2^0 = 1, and (2^0)^3 = 2^(0x3) = 2^0 = 1. The calculator correctly handles all zero-exponent cases.