Exponents and Roots: Chapter Review

Vocabulary and notation

Base and exponent

In $a^{n}$, the base $a$ is the number used as a factor and the exponent $n$ counts the copies multiplied. The exponent is a count, never a factor itself.

Squared, cubed

$a^{2}$ is "$a$ squared", the area of a square of side $a$; $a^{3}$ is "$a$ cubed", the volume of a cube of edge $a$.

Perfect square

A whole number reached by squaring a whole number: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$.

Radical sign $\sqrt{\phantom{x}}$ and radicand

The radical sign asks for a square root; the radicand is the number sitting under it.

Principal square root

The single non-negative value the radical returns. Both $6$ and $-6$ square to $36$, but $\sqrt{36} = 6$; the other is written $-\sqrt{36}$.

Irrational number

A number whose decimal neither ends nor repeats, so no fraction equals it. $\sqrt{N}$ is irrational whenever the whole number $N$ is not a perfect square, so $\sqrt{2}$ stays in radical form.

Reciprocal

Flipping a number over: $a$ gives $\tfrac{1}{a}$, and $\tfrac{2}{5}$ gives $\tfrac{5}{2}$. Only a nonzero number has one.

Coefficient and significant digits

In $a \times 10^{n}$ the coefficient $a$ carries the meaningful digits and $10^{n}$ carries the magnitude: $4.2 \times 10^{9}$ and $4.2 \times 10^{-9}$ differ only in size.

Renormalizing

Shifting a factor of ten between the coefficient and the power until the coefficient sits in $1 \le a < 10$: $0.4 \times 10^{6}$ becomes $4 \times 10^{5}$.

Formulas and theorems

• Definition of a power

  $$a^{n} = \underbrace{a \times a \times \cdots \times a}_{n \text{ copies of } a}$$

  Use when $n$ a whole number with $n \ge 1$; any base. $a^{1} = a$, so a number with no written exponent carries an exponent of $1$.

• Powers of ten

  $$10^{n} = 1\underbrace{00\ldots0}_{n \text{ zeros}}$$

  Use when $n$ a whole number with $n \ge 0$; the exponent is exactly the zero count, and for negative $n$ take the reciprocal instead. The exponent is the count; the power is the number it stands for.

• Exponents in the order of operations

  Parentheses, then exponents, then $\times$ and $\div$, then $+$ and $-$.

  Use when A power binds to its own base alone unless parentheses widen it: in $5 \times 2^{3}$ the exponent acts on the $2$, not the product.

• Sign of a power

  $$\begin{gathered} (-a)^{n} > 0 \text{ for even } n \\ (-a)^{n} < 0 \text{ for odd } n \\ -a^{n} = -(a^{n}) \end{gathered}$$

  Use when $a > 0$. Only parentheses pull the minus into the base; without them the power comes first and the sign after, leaving $-a^{n}$ negative for every $n$.

  e.g. $(-3)^{2} = 9$, $(-3)^{3} = -27$, and $-3^{2} = -9$.

• Product rule

  $$a^{m} \times a^{n} = a^{m+n}$$

  Use when The SAME base in both factors, and $a \neq 0$ once either exponent is zero or negative. Unlike bases never merge under this rule.

  e.g. $7^{3} \times 7^{2} = 7^{5}$.

• Quotient rule

  $$\frac{a^{m}}{a^{n}} = a^{m-n}$$

  Use when Same base top and bottom, and $a \neq 0$. Always subtract the denominator's exponent from the numerator's; the result may be zero or negative.

  e.g. $\dfrac{5^{8}}{5^{3}} = 5^{5}$, while $\dfrac{5^{3}}{5^{8}} = 5^{-5}$.

• Power rule

  $$\left(a^{m}\right)^{n} = a^{m \times n}$$

  Use when Any base with integer exponents, and $a \neq 0$ once an exponent is zero or negative.

  e.g. $\left(4^{3}\right)^{2} = 4^{6} = 4096$.

• Power of a product or a quotient

  $$(ab)^{n} = a^{n} b^{n}, \qquad \left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}$$

  Use when $b \neq 0$ for the quotient, and $a \neq 0$ too once $n$ is zero or negative. The exponent lands on each factor of a PRODUCT, never on the terms of a sum.

  e.g. $(2 \times 5)^{3} = 2^{3} \times 5^{3} = 8 \times 125 = 1000$.

• Zero exponent

  $$a^{0} = 1$$

  Use when $a \neq 0$, and $0^{0}$ is undefined. Every other base gives $1$, whole, fractional, or negative.

  e.g. $5^{0} = 1$, $\left(\tfrac{3}{7}\right)^{0} = 1$, and $(-4)^{0} = 1$.

• Negative exponent

  $$a^{-n} = \frac{1}{a^{n}}, \qquad \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$$

  Use when $a \neq 0$, and $b \neq 0$ for the fraction form. The minus sign moves the power across the fraction bar and never touches the sign of the value.

  e.g. $4^{-2} = \tfrac{1}{16}$ and $\left(\tfrac{3}{4}\right)^{-2} = \tfrac{16}{9}$.

• Square root as the inverse of squaring

  $$\sqrt{a^{2}} = a, \qquad \left(\sqrt{a}\right)^{2} = a$$

  Use when $a \ge 0$. The radical returns the principal (non-negative) root, one value and never $\pm$. A negative radicand has no real square root.

  e.g. $\sqrt{11^{2}} = 11$ and $\left(\sqrt{11}\right)^{2} = 11$.

• Product and quotient rules for square roots

  $$\sqrt{ab} = \sqrt{a}\,\sqrt{b}, \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$

  Use when $a \ge 0$ and $b \ge 0$, tightened to $b > 0$ for the quotient. Products and quotients split; sums and differences do not.

  e.g. $\sqrt{9 \times 49} = 3 \times 7 = 21$.

• Trapping a root between whole numbers

  $$p^{2} < N < q^{2} \;\Longrightarrow\; p < \sqrt{N} < q$$

  Text description

  Two aligned number lines: 60 lies between 49 and 64 on the upper one, and directly below, the square root of 60 lies between 7 and 8, nearer 8.

  Use when $N$, $p$, $q$ non-negative, with $q = p + 1$ for consecutive whole numbers. The root leans toward whichever perfect square $N$ is nearer.

  e.g. $49 < 60 < 64$, so $7 < \sqrt{60} < 8$, and nearer to $8$.

• Scientific notation

  $$N = a \times 10^{n}, \qquad 1 \le a < 10$$

  Text description

  The decimal point travels four places left to turn 45000 into 4.5 times ten to the fourth, and four places right to turn 0.00056 into 5.6 times ten to the negative fourth.

  Use when $N$ positive and $n$ an integer. Exactly one pair $(a, n)$ fits each $N$, which makes the form unique; $30 \times 10^{2}$ and $0.3 \times 10^{4}$ are equal in value but not in this form.

• Arithmetic in scientific notation

  $$\begin{gathered} \left(a \times 10^{m}\right)\left(b \times 10^{n}\right) \\ = ab \times 10^{m+n} \\ \frac{a \times 10^{m}}{b \times 10^{n}} = \frac{a}{b} \times 10^{m-n} \end{gathered}$$

  Use when $b \neq 0$ for the quotient. Neither result is finished until renormalized, since $ab$ or $\tfrac{a}{b}$ can leave $1 \le a < 10$.

• Comparing two numbers in scientific notation

  $$m > n \;\Longrightarrow\; a \times 10^{m} > b \times 10^{n}$$

  Use when Both positive and normalized, $1 \le a < 10$ and $1 \le b < 10$; an inflated coefficient breaks it. Coefficients decide only when the exponents tie.

  e.g. $3.5 \times 10^{6} > 8.1 \times 10^{4}$, despite the smaller coefficient.

Problem types, step by step

Exam traps

• Trap Multiplying the base by the exponent, so $2^{5}$ is answered as $10$.

  Fix The exponent counts factors: $2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32$.

• Trap Multiplying the exponents, or the bases, when two powers merely sit side by side: $2^{3} \times 2^{4}$ answered as $2^{12}$ or $4^{7}$.

  Fix Side by side the exponents ADD and the base is untouched: $2^{3} \times 2^{4} = 2^{7} = 128$. Exponents multiply only for a power of a power, $\left(2^{3}\right)^{4} = 2^{12}$.

• Trap Combining powers of unlike bases, so $2^{3} \times 3^{2}$ becomes $6^{5}$.

  Fix Every product and quotient law needs the SAME base. $2^{3} \times 3^{2} = 8 \times 9 = 72$, whereas $6^{5} = 7776$.

• Trap Using an exponent law on a SUM of powers, so $2^{3} + 2^{4}$ becomes $2^{7}$.

  Fix The laws cover multiplying and dividing only: $2^{3} + 2^{4} = 8 + 16 = 24$, while $2^{7} = 128$.

• Trap Splitting a power or a radical across a sum: $(2 + 3)^{2}$ read as $2^{2} + 3^{2}$, or $\sqrt{9 + 16}$ read as $3 + 4$.

  Fix Combine under the grouping first: $(2 + 3)^{2} = 25$, not $13$, and $\sqrt{9 + 16} = \sqrt{25} = 5$, not $7$. Only products and quotients split.

• Trap Reading $a^{0}$ as $0$, so $9^{0}$ is answered as $0$.

  Fix A zero exponent means no factors of the base, which leaves the multiplication at $1$: $9^{0} = 1$. Only $0^{0}$ is undefined.

• Trap Treating a negative exponent as a negative value, so $2^{-3}$ is answered as $-8$.

  Fix It calls for a reciprocal, not a sign change: $2^{-3} = \tfrac{1}{2^{3}} = \tfrac{1}{8}$. A value goes negative only from a negative BASE with an odd exponent, as in $(-2)^{-3} = -\tfrac{1}{8}$.

• Trap Letting a bare minus sign into the base, so $-4^{2}$ is answered as $16$.

  Fix Only parentheses capture the sign: $-4^{2} = -(4 \times 4) = -16$, while $(-4)^{2} = 16$.

• Trap Answering $\sqrt{16}$ with $-4$ or with $\pm 4$.

  Fix The radical returns the principal root alone, $\sqrt{16} = 4$. The negative solution must be asked for, as $-\sqrt{16}$.

• Trap Stopping at $18 \times 10^{8}$ or $0.5 \times 10^{3}$ and calling it scientific notation.

  Fix Renormalize until $1 \le a < 10$: $18 \times 10^{8} = 1.8 \times 10^{9}$, and $0.5 \times 10^{3} = 5 \times 10^{2}$.