Math & Scientific
Identify a Base and Exponent or Recover a Missing One
Read power notation, distinguish coefficients and signs, and recover an unknown base or exponent with roots or logarithms under stated conditions.
Read the notation before trying to solve anything
In 7^3, the base is seven and the exponent is three. In 4×7^3, four is an outside coefficient; it is not part of the base unless parentheses specify (4×7)^3. The first expression gives 4×343 = 1372, while the grouped version gives 28^3 = 21952. Write the base as the complete quantity governed by the superscript. A fraction, negative number or parenthesized expression can be a base just as a positive integer can.
Keep a leading minus sign in its correct place
For (−5)^2, the base is −5 and the exponent is two, giving 25. For −(5^2), the base of the power is five and the overall expression is −25. A worksheet may omit parentheses around a positive base but needs clear grouping when signs can be read differently. The local base field takes the complete signed numeric base, so entering −5 and exponent 2 checks the first interpretation. It does not automatically insert an additional external minus sign.
Recover a positive base from a known power
If b^3 = 125, taking the real cube root gives b = 5. Enter base 5 and exponent 3 to verify 125. For b^4 = 81, both b = 3 and b = −3 are real solutions because an even power removes the sign. If the problem explicitly requires a positive base, choose 3; otherwise report both real candidates. A square-root symbol normally denotes the principal nonnegative root, while solving a squared equation can require a plus-or-minus pair.
Recover an exponent by matching a familiar power
For 2^x = 32, repeated powers 2, 4, 8, 16 and 32 give x = 5. For 10^x = 0.001, recognize 0.001 = 1/1000 = 10^(−3), so x = −3. These exact patterns avoid unnecessary decimal approximation. Entering base 2 and exponent 5, or base 10 and exponent −3, verifies the result. The calculator evaluates the proposed pair; it does not search the unknown x from an equation supplied in a single field.
Use logarithms when the exponent is not obvious
For b^x = y with b positive, b different from one and y positive, x = ln(y)/ln(b). If 3^x = 20, x is approximately 2.726833. It lies between two and three because 3^2 = 9 and 3^3 = 27. This bound is a useful check on the logarithm calculation. Negative or zero bases and nonpositive target values do not satisfy these real-logarithm conditions and require separate analysis rather than substitution into the same quotient.
Recognize cases that do not identify a unique answer
If 1^x = 1, many exponents satisfy the equation, so the value one does not determine x. If b^0 = 1 with b nonzero, the equation also does not determine a unique base. Knowing only the result 64 leaves many base-exponent pairs, such as 2^6, 4^3 and 8^2. Additional information about one quantity or the permitted domain is required. A single numerical result is not enough to infer one uniquely correct base and exponent pair.
Distinguish a scientific-notation exponent
In 6.2×10^4, ten is the base of the power, four is its exponent and 6.2 is the coefficient. The complete value is 62000. Calling 6.2 the base of that power would confuse the coefficient with the repeated factor. Scientific notation usually fixes base ten and chooses a coefficient with magnitude from one up to but below ten. This convention answers a representation question; it is different from solving an arbitrary exponential equation for its base or exponent.
Verify a recovered candidate in the original equation
For b³ = −64, the real candidate is b = −4, and (−4)³ returns −64. Testing a proposed answer in the original equation checks its sign as well as its magnitude. A candidate obtained by taking an absolute value too early would fail that check.
Calculator for the supported step
The Exponent Calculator takes a known base and exponent and checks their power. Recovering an unknown base or exponent is an algebraic step that must be completed before using this form.
Sources and method
Related guides
Evaluate powers, distinguish an exponent from a multiplier, apply exponent rules, and handle negative bases and fractional exponents deliberately.
Find square and higher roots, distinguish principal roots from equation solutions, and handle negative radicands, exact forms and numerical checks.
Convert large and small decimals to scientific notation, choose the exponent sign, interpret E notation, and check rounding that changes the exponent.