Expressions, Functions & Data · 7 min read
Exponents and Radicals
This reporting category opens with number sense: rewriting expressions that contain exponents and radicals. On the test you may be asked to choose an expression equivalent to a given one, to evaluate a number raised to a fraction exponent, or to carry out one operation on two square roots or two cube roots. Every problem in this chapter is solved with a small set of rules applied carefully.
Laws of Exponents
An exponent counts how many times the base is used as a factor: ~x^{4} = x · x · x · x~. The laws in the table follow from that meaning. They hold for every integer exponent and, when the bases are positive, for fraction exponents as well. The product and quotient laws apply only to powers of the same base.
| Law | Rule | Example |
|---|---|---|
| Product | ~a^{m} · a^{n} = a^{m + n}~ | ~x^{3} · x^{5} = x^{8}~ |
| Quotient | ~a^{m} ÷ a^{n} = a^{m − n}~, where ~a ≠ 0~ | ~y^{7} ÷ y^{2} = y^{5}~ |
| Power of a power | ~(a^{m})^{n} = a^{mn}~ | ~(z^{2})^{4} = z^{8}~ |
| Power of a product | ~(ab)^{n} = a^{n}b^{n}~ | ~(2x)^{3} = 2^{3}x^{3} = 8x^{3}~ |
| Power of a quotient | ~(a ÷ b)^{n} = a^{n} ÷ b^{n}~, where ~b ≠ 0~ | ~(x ÷ 3)^{2} = x^{2} ÷ 9~ |
| Zero exponent | ~a^{0} = 1~, where ~a ≠ 0~ | ~5^{0} = 1~ and ~(−7)^{0} = 1~ |
| Negative exponent | ~a^{−n} = [[1//a^{n}]]~, where ~a ≠ 0~ | ~2^{−3} = [[1//2^{3}]] = [[1//8]]~ |
- Raise every factor in the first group to the 4th power: ~2^{4}x^{12}y^{8} = 16x^{12}y^{8}~.
- Raise every factor in the second group to the 2nd power: ~3^{2}x^{4}y^{2} = 9x^{4}y^{2}~.
- Multiply the coefficients and add the exponents of like bases: ~16 · 9 = 144~, ~x^{12 + 4} = x^{16}~ and ~y^{8 + 2} = y^{10}~.
- Answer: ~144x^{16}y^{10}~
- First expression: divide the coefficients, ~24 ÷ 8 = 3~, and subtract the exponents of like bases, ~a^{5 − 2} = a^{3}~ and ~b^{3 − 7} = b^{−4}~.
- Rewrite the negative exponent as a reciprocal: ~3a^{3}b^{−4} = [[3a^{3}//b^{4}]]~.
- Second expression: apply the outside exponent to every factor, ~3^{−2} · x^{(−2)(−2)} · y^{(4)(−2)} = 3^{−2}x^{4}y^{−8}~.
- Move the factors with negative exponents to the denominator: ~3^{−2} = [[1//9]]~ and ~y^{−8} = [[1//y^{8}]]~.
- Answer: ~[[3a^{3}//b^{4}]]~ and ~[[x^{4}//9y^{8}]]~
Rational Exponents
A rational (fraction) exponent is another way to write a root. The denominator of the exponent is the index of the root and the numerator is a power. Because ~(a^{1/2})^{2} = a^{1} = a~, the number ~a^{1/2}~ must be the square root of ~a~. The rules in the box assume a positive base ~a~.
- ~a^{1/n} = ⁿ√a~, so ~a^{1/2} = √a~ and ~a^{1/3} = ³√a~.
- ~a^{m/n} = (ⁿ√a)^{m} = ⁿ√{a^{m}}~.
- Converting between forms: ~³√{5^{2}} = 5^{2/3}~, ~(√7)^{3} = 7^{3/2}~ and ~10^{3/4} = ⁴√{10^{3}}~.
- To evaluate, take the root first whenever the base is a perfect power, so that the numbers stay small.
- ~27^{2/3} = (³√{27})^{2} = 3^{2} = 9~
- ~16^{3/4} = (⁴√{16})^{3} = 2^{3} = 8~
- ~8^{−2/3} = [[1//8^{2/3}]] = [[1//(³√8)^{2}]] = [[1//2^{2}]] = [[1//4]]~
- Answer: 9, 8 and ~[[1//4]]~
- Product law: ~x^{1/2} · x^{3/2} = x^{1/2 + 3/2} = x^{2}~.
- Power of a product: ~(16x^{4})^{1/2} = 16^{1/2} · x^{4 · 1/2} = 4x^{2}~.
- Power of a product: ~(8x^{6})^{2/3} = 8^{2/3} · x^{6 · 2/3} = 4x^{4}~, because ~8^{2/3} = (³√8)^{2} = 2^{2} = 4~.
- Answer: ~x^{2}~, ~4x^{2}~ and ~4x^{4}~
Simplifying Square Roots and Cube Roots
A square root is in simplest form when the number under the radical sign (the radicand) has no perfect-square factor other than 1. A cube root is in simplest form when the radicand has no perfect-cube factor other than 1. To simplify, split the radicand into its largest perfect-power factor and what is left, using the product property ~√{ab} = √a · √b~. The cube root of a negative number is negative, as in ~³√{−8} = −2~; the square root of a negative number is not a real number.
- The largest perfect-square factor of 72 is 36: ~√{72} = √{36} · √2 = 6√2~.
- The largest perfect-cube factor of 54 is 27: ~³√{54} = ³√{27} · ³√2 = 3 · ³√2~.
- In the same way, ~√{50} = √{25} · √2 = 5√2~, ~√{200} = √{100} · √2 = 10√2~ and ~³√{16} = ³√8 · ³√2 = 2 · ³√2~.
- Answer: ~6√2~ and ~3 · ³√2~
Adding and Subtracting Radicals
Radicals combine the way like terms do. Like radicals have the same index and the same radicand, such as ~3√5~ and ~7√5~. Add or subtract the numbers in front and keep the radical: ~3√5 + 7√5 = 10√5~. Unlike radicals such as ~2√3 + 4√5~ cannot be combined. Simplify each radical first, because radicals that look different may turn out to be alike.
- ~√{12} = 2√3~ and ~√{27} = 3√3~, so ~√{12} + √{27} = 2√3 + 3√3 = 5√3~.
- ~√{50} = 5√2~ and ~√{18} = 3√2~, so ~√{50} − √{18} = 5√2 − 3√2 = 2√2~.
- ~³√{16} = 2 · ³√2~ and ~³√{54} = 3 · ³√2~, so the sum is ~5 · ³√2~.
- Answer: ~5√3~, ~2√2~ and ~5 · ³√2~
Multiplying and Dividing Radicals
To multiply radicals with the same index, multiply the numbers in front, multiply the radicands, then simplify: ~√3 · √7 = √{21}~ and ~√{12} · √3 = √{36} = 6~. Squaring a square root removes the radical, so ~(2√3)^{2} = 2^{2} · (√3)^{2} = 4 · 3 = 12~. To divide, divide the numbers in front and divide the radicands, using ~√a ÷ √b = √{a ÷ b}~.
- Multiply the outside numbers and the radicands, then simplify: ~2√5 · 3√{10} = 6√{50} = 6 · 5√2 = 30√2~.
- For the quotient, ~6 ÷ 2 = 3~ and ~√{15} ÷ √3 = √5~, which gives ~3√5~.
- Cube roots follow the same pattern: ~³√4 · ³√6 = ³√{24} = ³√8 · ³√3 = 2 · ³√3~, and ~³√{54} ÷ ³√2 = ³√{27} = 3~.
- Answer: ~30√2~ and ~3√5~
Rationalizing a Denominator
An answer in simplest form has no radical in the denominator. To rationalize a denominator that is a single square root, multiply the numerator and the denominator by that square root. This multiplies the fraction by 1, so its value does not change, and ~√a · √a = a~ clears the radical.
- ~[[3//√5]] · [[√5//√5]] = [[3√5//5]]~
- ~[[6//√2]] · [[√2//√2]] = [[6√2//2]] = 3√2~
- Simplify first: ~√{12} = 2√3~, so ~[[2//√{12}]] = [[2//2√3]] = [[1//√3]]~. Then ~[[1//√3]] · [[√3//√3]] = [[√3//3]]~.
- Answer: ~[[3√5//5]]~, ~3√2~ and ~[[√3//3]]~
- Same base: multiply → add exponents; divide → subtract exponents; power of a power → multiply exponents.
- ~a^{0} = 1~ and ~a^{−n} = [[1//a^{n}]]~ when ~a ≠ 0~. A negative exponent means a reciprocal, never a negative number.
- ~a^{m/n} = (ⁿ√a)^{m}~: the denominator is the root and the numerator is the power.
- Simplify a radical by pulling out the largest perfect-square (or perfect-cube) factor.
- Only like radicals can be added or subtracted; radicands are multiplied or divided under one radical.
- Rationalize ~[[a//√b]]~ by multiplying the numerator and denominator by ~√b~.
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