Rewrite Using a Single Exponent Calculator

A single-exponent calculator rewrites a product of powers with the same base as one power, using the sum of their exponents. Enter the exponents you want to combine, and the calculator returns their sum; the base stays unchanged.

Formula

For powers with the same nonzero base, multiply by adding the exponents:

x^a × x^b = x^{a + b}

• x is the shared base, a number or variable. It is unchanged by this rewrite.
• a and b are the exponents, which are numbers without units in ordinary algebra.
• n = a + b is the combined exponent returned by the calculator.

The calculator accepts two exponents and displays their sum. The base is not included as an input because the same-base multiplication rule does not change it. For example, if the base is 5, the inputs 2 and 3 give 5⁵.

Worked Example

Using the calculator’s defaults, the exponents are a = 2 and b = 3:

1. Start with x² × x³.
2. Add the exponents: 2 + 3 = 5.
3. Keep the common base and write x⁵.

The calculator’s combined exponent is 5. As a numerical check with x = 2, 2² × 2³ = 4 × 8 = 32, and 2⁵ = 32.

How Rewriting With One Exponent Works

An exponent tells how many times a base is used as a factor when the exponent is a positive whole number. Thus, x² × x³ means (x × x) × (x × x × x), which contains five factors of x. That is why the product is x⁵.

The same rule works for more than two powers. Add every exponent attached to the identical base:

• y⁴ × y⁶ = y¹⁰
• 3² × 3⁵ × 3¹ = 3⁸
• m × m⁷ = m⁸, because an unwritten exponent of 1 is understood on m.

For several factors, add the exponents one at a time or sum them together. The result is the same, provided the base is common and the operation is multiplication.

When to Use the Rule

Use this rewrite when multiplying powers that have exactly the same base. It is useful for simplifying algebraic expressions, combining repeated factors, and making expressions easier to compare or calculate. It applies to numerical bases such as 10 and to variables such as x or r.

For instance, 10³ × 10⁴ becomes 10⁷. This does not mean 10³ + 10⁴ becomes 10⁷: the exponent rule applies to multiplication of powers, not addition of their values.

Common Mistakes and Limits

• Combining unlike bases: 2³ × 3⁴ cannot be rewritten as one power by adding 3 and 4. The bases differ. You may evaluate the powers or use other algebraic methods, but this rule does not apply directly.
• Adding powers instead of multiplying: x² + x³ is generally not x⁵. For example, 2² + 2³ = 12, not 32.
• Using the multiplication rule for division: When dividing powers with the same nonzero base, subtract exponents: x^a ÷ x^b = x^{a − b}. Do not add them.
• Confusing powers of powers: For (x^a)^b, multiply exponents: x^ab. This differs from multiplying two powers with the same base.
• Forgetting exponent 1: A bare variable such as x is x¹, so x × x⁴ = x⁵.

The stated product rule is safe for ordinary real-number algebra when the powers are defined; treating zero or negative bases with fractional exponents can require additional conventions, especially when working only with real numbers. For routine integer-exponent problems, the key check is simple: confirm the bases match and the operation is multiplication.

The calculator adds the exponent values exactly as entered, then displays a whole-number result because its inputs are set to whole numbers. If you are working with fractional exponents, add them as fractions or decimals and retain suitable precision. Rounding the exponent too early can change the value of the expression.

Frequently Asked Questions

How do I combine powers with the same base?
When multiplying them, keep the base and add the exponents: a^m × aⁿ = a^m+n.

Do you add exponents when dividing powers?
No. For the same nonzero base, subtract the denominator’s exponent from the numerator’s: a^m ÷ aⁿ = a^m−n.

Can I combine powers with different bases?
Not using the same-base rule. For example, 2³ × 3² cannot be turned into a single power by adding the exponents.

What if one factor has no written exponent?
A base by itself has exponent 1. Therefore, x × x⁵ = x⁶.

Is x² + x³ equal to x⁵?
No. Adding powers is different from multiplying them, so the exponents cannot be combined by this rule.

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in a | First exponent | - | 2 | 0
in b | Second exponent | - | 3 | 0
out n | Combined exponent | - | a + b | 0