Exponents, Exponentials and Logarithms · Lesson 1

Exponent rules

Every exponent rule comes from counting factors. When you multiply powers of the same base, add the exponents: xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}. When you divide, subtract: xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}. A power of a power multiplies: (xa)b=xab(x^a)^b = x^{ab}. A power of a product reaches every factor, coefficient included: (2y5)3=8y15(2y^5)^3 = 8y^{15}.

Zero and negative exponents. For x≠0x \neq 0, x0=1x^0 = 1 and x−n=1xnx^{-n} = \frac{1}{x^n}. A negative exponent moves the base to the other side of the fraction bar; it never makes the value negative. So 2−3=182^{-3} = \frac{1}{8} and (25)−2=254\left(\frac{2}{5}\right)^{-2} = \frac{25}{4}.

Fractional exponents are roots. am/n=(an)ma^{m/n} = (\sqrt[n]{a})^m: the denominator is the root and the numerator is the power. Take the root first so the numbers stay small: 323/5=(325)3=23=832^{3/5} = (\sqrt[5]{32})^3 = 2^3 = 8.

Undo a fractional exponent with its reciprocal. If x3/4=27x^{3/4} = 27 and x>0x > 0, raise both sides to the 43\frac{4}{3} power: x=274/3=(273)4=34=81x = 27^{4/3} = (\sqrt[3]{27})^4 = 3^4 = 81.

The exponent rules, with x≠0x \neq 0
RuleExampleResult
xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}x5⋅x−2x^5 \cdot x^{-2}x3x^3
xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}x4x7\frac{x^4}{x^7}x−3=1x3x^{-3} = \frac{1}{x^3}
(xa)b=xab(x^a)^b = x^{ab}(x−2)3(x^{-2})^3x−6x^{-6}
am/n=(an)ma^{m/n} = (\sqrt[n]{a})^m163/416^{3/4}23=82^3 = 8

Worked example Easy

What is the value of 27−2/327^{-2/3}?

  1. −9-9
  2. 19\frac{1}{9}Answer
  3. 13\frac{1}{3}
  4. 99

How to solve it

  1. The negative exponent means a reciprocal: 27−2/3=1272/327^{-2/3} = \frac{1}{27^{2/3}}.
  2. The denominator 3 means a cube root: 273=3\sqrt[3]{27} = 3.
  3. The numerator 2 means a square: 32=93^2 = 9, so 272/3=927^{2/3} = 9.
  4. So 27−2/3=1927^{-2/3} = \frac{1}{9}.

Why each choice is right or wrong

  • A. Incorrect. −9-9 treats the negative exponent as a negative sign. A negative exponent takes the reciprocal, so the value stays positive.
  • B. Correct. 272/3=(273)2=32=927^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9, and the negative exponent flips it: 27−2/3=1927^{-2/3} = \frac{1}{9}.
  • C. Incorrect. 13\frac{1}{3} stops one step early: it is 27−1/327^{-1/3}. The numerator 2 still squares the cube root, so the value is 132\frac{1}{3^2}.
  • D. Incorrect. 9 is 272/327^{2/3}, which drops the negative sign of the exponent. That sign means take the reciprocal of 9.

Worked example Medium

For nonzero xx and yy, which expression is equivalent to (2x3y−1)48x5y2\frac{(2x^3y^{-1})^4}{8x^5y^2}?

  1. x74y6\frac{x^7}{4y^6}
  2. 2x2y6\frac{2x^2}{y^6}
  3. 2x7y2\frac{2x^7}{y^2}
  4. 2x7y6\frac{2x^7}{y^6}Answer

How to solve it

  1. Raise every factor in the parentheses to the 4th power: 24=162^4 = 16, (x3)4=x12(x^3)^4 = x^{12}, (y−1)4=y−4(y^{-1})^4 = y^{-4}.
  2. Divide the coefficients: 168=2\frac{16}{8} = 2.
  3. Subtract exponents: x12−5=x7x^{12-5} = x^7 and y−4−2=y−6y^{-4-2} = y^{-6}.
  4. Move y−6y^{-6} to the denominator: 2x7y6\frac{2x^7}{y^6}.

Why each choice is right or wrong

  • A. Incorrect. x74y6\frac{x^7}{4y^6} leaves the coefficient 2 unraised and uses 28=14\frac{2}{8} = \frac{1}{4}. The power 4 applies to every factor inside the parentheses, so the 2 becomes 16.
  • B. Incorrect. 2x2y6\frac{2x^2}{y^6} adds the exponents 3 and 4 to get x7x^7 in the numerator. A power of a power multiplies: (x3)4=x12(x^3)^4 = x^{12}, and x12÷x5=x7x^{12} \div x^5 = x^7.
  • C. Incorrect. 2x7y2\frac{2x^7}{y^2} comes from a sign slip, y−4÷y2=y−2y^{-4} \div y^2 = y^{-2}. Subtracting gives −4−2=−6-4 - 2 = -6, so the yy part is 1y6\frac{1}{y^6}.
  • D. Correct. The numerator is 16x12y−416x^{12}y^{-4}. Dividing by 8x5y28x^5y^2 gives 2x7y−6=2x7y62x^7y^{-6} = \frac{2x^7}{y^6}.

Trap The coefficient gets the power too

(2x3)2(2x^3)^2 is 4x64x^6, not 2x62x^6: the outside exponent applies to the 2 as well as to x3x^3. The same goes for a negative coefficient: (−3x)2=9x2(-3x)^2 = 9x^2. Choices that forget to raise the coefficient are common.

Trap A negative exponent is not a negative number

5−2=1255^{-2} = \frac{1}{25}, a small positive number. Choices such as −25-25 or −125-\frac{1}{25} are there for anyone who reads the minus sign in the exponent as the sign of the value.

Calculator When a calculator isn’t faster

Hand work is usually quicker here: the choices are expressions, and the rules take a few seconds each. To check an equivalence, graph the original and a choice in the built-in Desmos calculator if you test online; equivalent expressions draw the same graph.

For a number such as 27−2/327^{-2/3}, your calculator shows 0.1111..., which you then have to recognize as 19\frac{1}{9}. Knowing the root-then-power routine is faster.