Intro to Exponents
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Powers and Roots
Summary
Powers and roots begin with the idea that exponents are repeated multiplication, just as multiplication is repeated addition. On the ACT, the key skill is not computing large powers directly but understanding how exponents behave, especially with positive integer exponents, negative bases, and common powers you should know from memory.
- Multiplication condenses repeated addition, and exponents condense repeated multiplication; for example, b to the n means n factors of b multiplied together, with b as the base, n as the exponent, and the whole expression as a power.
- The focus stays on positive integer exponents, since that definition works cleanly there, while negative and fractional exponents are saved for later; exponents can apply to both numbers and variables, and common language includes squared for a power of 2 and cubed for a power of 3.
- Several foundational cases matter: 1 to any power is always 1, 0 to any positive power is 0, and when no exponent is written, the exponent is understood to be 1.
- Negative bases follow a sign pattern: an even exponent gives a positive result, and an odd exponent gives a negative result; this pattern helps explain why equations like x squared equals 4 have two solutions, while x cubed equals 8 has one real solution.
- Even powers cannot equal a negative number, so equations such as a squared expression equaling a negative have no solution, while odd powers can equal negatives.
- For ACT success, memorize common powers: powers of 2 through at least 2 to the 9th, powers of 3 through 3 to the 4th, powers of 4 and 5 through the 4th power, squares and cubes of 6 through 9, and powers of 10.
Chapters
00:00
Exponents as Repeated Multiplication
01:32
Positive Integer Exponents and Basic Terminology
04:44
Negative Bases and Solving Power Equations
06:27
Essential Powers to Memorize for the ACT