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Exponents & Radicals Lab

Explore how exponent rules simplify expressions. Apply the product, quotient, power, negative, and zero exponent rules interactively. See how rational exponents connect to radicals and verify each step.

Guided Experiment: Exponent Rules Exploration

How do exponent rules (product, quotient, power) simplify expressions with the same base? What happens with zero and negative exponents?

Write your hypothesis in the Lab Report panel, then click Next.

Exponential Growth

Value12^022^142^282^3162^4322^5642^6Powers of 2

Controls

First expression: base^exponent
×
Second expression: base^exponent

Simplification Steps

1Start with the expression
23⋅242^{3} \cdot 2^{4}
2Apply the product rule: a^m · a^n = a^(m+n)Product Rule
272^{7}
3Evaluate
=128= 128
Result
272^{7}
= 128
Product Rule

Data Table

(0 rows)
#ExpressionSimplifiedRule AppliedDecimal Value
0 / 500
0 / 500
0 / 500

Reference Guide

Product & Quotient Rules

When multiplying powers with the same base, add the exponents. When dividing, subtract them.

am⋅an=am+na^m \cdot a^n = a^{m+n}
aman=am−n\frac{a^m}{a^n} = a^{m-n}

These rules work because exponents count repeated multiplication.

Power & Zero Rules

Raising a power to another power multiplies the exponents. Any nonzero base to the zero power equals 1.

(am)n=amn(a^m)^n = a^{mn}
a0=1(a≠0)a^0 = 1 \quad (a \ne 0)

The zero exponent rule follows from the quotient rule when m = n.

Negative Exponents

A negative exponent means "take the reciprocal." The base moves to the other side of the fraction bar.

a−n=1ana^{-n} = \frac{1}{a^n}
5−2=152=1255^{-2} = \frac{1}{5^2} = \frac{1}{25}

This rule follows from the quotient rule when the numerator exponent is smaller.

Rational Exponents & Radicals

A fractional exponent connects powers and roots. The denominator is the root index, the numerator is the power.

amn=amn=(an)ma^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m
823=823=643=48^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4

You can compute the root first or the power first and get the same answer.