A power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.
In Spanish
potencia
Also called
exponential expression, raised to a power, nth power, powers
A power multiplies the base by itself again and again. Going up one power multiplies by the base once more.
Formal definition and notation
For a numberb and a positive whole numbern, the nth power of b is bn = b × b × ⋯ × b with n factors; b is the base and n is the exponent. The expression bn and its value are both called a power, and some books also call the exponent itself the power. The definition extends to b0 = 1 for b ≠ 0 and to negative and fractional exponents.
Symbol
Read it as
Meaning
bn
b to the nth power, or b to the n
n factors of b multiplied together
b2
b squared
the second power: the area of a square with side b
The exponent is the small raised number (3 in 53). The power is the whole expression 53 or its value, 125, though some books use the words interchangeably.
The base is the number being multiplied (5 in 53); the power is the result of multiplying it.
Quick check: 3 questions
Question 1 of 3
What is the third power of 4?
Show answer and explanation
Answer: B.43 = 4 × 4 × 4 = 64. 12 multiplies 4 by 3, 16 is the second power, and 7 adds 4 and 3.
Question 2 of 3
Which number is a power of 10?
Show answer and explanation
Answer: A.100,000 = 105, a 1 followed by five zeros. 10,010 has a 1 in two places, 500 is 5 × 102, and 20 is 2 × 10.
Question 3 of 3
Which list shows the first four powers of 3 (from 31)?
Show answer and explanation
Answer: C.31 = 3, 32 = 9, 33 = 27, 34 = 81. A lists multiples of 3, B starts at 30, and D doubles after 9 instead of tripling.
Power: frequently asked questions
What is the difference between a power and an exponent?
The exponent is the small number that says how many times the base is used as a factor. The power is the whole expression, or the number it equals. In 24 = 16, the exponent is 4 and 16 is the fourth power of 2. In everyday speech people often blur the two, saying “2 to the power 4” for the exponent.
How do you say powers out loud?
Say the base, then “to the”, then the exponent as an ordinal, then “power”: 75 is “seven to the fifth power” or just “seven to the fifth”. The second and third powers have special names: 72 is “seven squared” and 73 is “seven cubed”. xn is read “x to the n”.
Why is any number to the power of 0 equal to 1?
Follow the pattern down the powers of a number: 23 = 8, 22 = 4, 21 = 2. Each step down divides by 2, so the next step is 2 ÷ 2 = 1, which makes 20 = 1. The same pattern works for any nonzero base. 00 is left undefined in most school math.
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