Mathematics KnowledgeLesson 7 of 21
Exponents and Roots
Exponent rules, perfect squares, and simplifying radicals.
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Exponent questions are fast points if you know the rules and guaranteed losses if you half-know them, because every wrong rule produces a clean-looking number that is sitting in the choices.
The exponent rules
| Rule | Form | Example |
|---|---|---|
| Multiply like bases | add the exponents | x cubed times x to the fourth is x to the seventh |
| Divide like bases | subtract the exponents | x to the fifth over x squared is x cubed |
| Power of a power | multiply the exponents | (x squared) cubed is x to the sixth |
| Power of a product | applies to each factor | (2x) cubed is 8x cubed |
| Zero exponent | equals 1 | 7 to the zero is 1 |
| Negative exponent | take the reciprocal | x to the -2 is 1 over x squared |
Two of those need emphasis.
Multiplying adds, and a power of a power multiplies. Mixing them up is the main error in the topic. x squared times x cubed is x to the fifth, but (x squared) cubed is x to the sixth. Both 5 and 6 will be choices.
The exponent rules need the same base. 2 cubed times 2 squared is 2 to the fifth. 2 cubed times 3 squared is nothing simpler than 8 times 9, which is 72. There is no rule for unlike bases, and this is asked.
Zero and negative exponents
Anything nonzero to the power 0 is 1. Not 0. This follows from the division rule: x cubed over x cubed is x to the zero, and it is also obviously 1.
A negative exponent is a reciprocal, not a negative value.
- 2 to the -3 is 1 over 2 cubed, which is 1/8. A positive number.
- x to the -1 is 1/x.
A negative exponent flips the base across the fraction bar and turns positive.
| Expression | Value |
|---|---|
| 5 to the 0 | 1 |
| 5 to the -1 | 1/5 |
| 5 to the -2 | 1/25 |
| (1/5) to the -2 | 25 |
The last row is the one worth a second look: flipping a fraction's base makes it larger.
Perfect squares and cubes
Know these on sight. Recognizing them is what makes radical work fast.
Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
That is 1 through 15 squared, and it is worth memorizing to 15 rather than to 12, because 13, 14 and 15 squared show up in Pythagorean and area questions.
Cubes: 1, 8, 27, 64, 125, 216.
Notice 64 is both a perfect square and a perfect cube.
Simplifying radicals
Pull out perfect-square factors.
Simplify the square root of 72.
Find the largest perfect square that divides 72. That is 36, since 72 is 36 times 2.
The square root of 72 is the square root of 36 times the square root of 2, which is 6 times the square root of 2.
If you do not spot the largest square, a smaller one still works and just takes another pass: 72 is 4 times 18, giving 2 times the root of 18; then 18 is 9 times 2, giving 2 times 3 times the root of 2, which is the same 6 root 2.
The rule that does not exist
The square root of a sum is not the sum of the square roots.
The square root of (9 plus 16) is the square root of 25, which is 5. It is not 3 plus 4. This is the most frequently asked misconception in the topic.
Roots do distribute over multiplication and division: the root of 4 times 9 is 2 times 3. They never distribute over addition or subtraction.
Operating on radicals
Adding radicals needs the same radicand, and then they behave like like terms.
- 3 root 5 plus 2 root 5 is 5 root 5.
- 3 root 5 plus 2 root 7 does not combine at all.
Multiplying radicals multiplies the insides.
- Root 6 times root 10 is root 60, which is root 4 times root 15, or 2 root 15.
A square root times itself returns the radicand: root 7 times root 7 is 7.
Every positive number has two square roots, one positive and one negative, since both 4 and -4 square to 16. The radical symbol means the positive one only, so the square root of 16 is 4. When you take the root of both sides while solving an equation, that is when both signs matter and you write plus-or-minus.
Fractional exponents
Lightly asked, and one rule covers it: the denominator of the exponent is the root.
- x to the 1/2 is the square root of x.
- 8 to the 1/3 is the cube root of 8, which is 2.
- x to the 2/3 is the cube root of x, squared.
What you can skip
Across the 144 questions on this topic:
- Rationalizing denominators never appears.
- Exponents beyond the rules. The zero exponent comes up once; fractional exponents and cube roots do appear, and are covered above.
Where people lose points
Multiplying exponents when multiplying bases. x squared times x cubed is x to the fifth, not x to the sixth.
Treating a zero exponent as 0. It is 1.
Treating a negative exponent as a negative number. It is a reciprocal.
Applying a rule across unlike bases.
Distributing a root over a sum.
Forgetting to apply an outer exponent to the coefficient. (3x) squared is 9x squared, not 3x squared.
Work one in under a minute
Simplify: (2x cubed)(3x to the fourth), then divide by 6x squared.
Numerator: multiply the coefficients, 2 times 3 is 6; add the exponents, 3 plus 4 is 7. So 6x to the seventh.
Divide: 6 over 6 is 1; subtract exponents, 7 minus 2 is 5.
Answer: x to the fifth.
Where this leads
Exponent rules govern scientific notation, and radical simplification is what the Pythagorean theorem hands you on most triangle questions.
Related lessonsReference
- Scientific Notation - the exponent rules applied to powers of ten
- Triangles and the Pythagorean Theorem - where radicals come out of a right triangle
- Factors, Multiples and Primes - finding the perfect-square factor to pull out
- Signed Numbers and Absolute Value - negative bases, and why parentheses decide the sign
Practice this topic
Check that this lesson stuck. Answer questions on exponents and roots only, and see the right answer and why after each one.
Practice Exponents and Roots questions