Digital SAT · Math · Unit 2 · Advanced Math

Lesson 2.2 · Exponents & Radicals

On the test: 1–2 questions a test; the hard ones are always “match the exponents”

Big idea. An exponent counts how many times a number is multiplied by itself. Every rule below is just counting those factors. A root is the reverse of a power, so a root can always be written as a FRACTION exponent — and then the same counting rules finish the question.

🧑‍🏫 Rule 1 · Multiply the same base: ADD the exponentsstep 1 of 4

1
count
$x^{2} \cdot x^{3} = (x \cdot x) (x \cdot x \cdot x) = x^{5}$
2 factors and 3 factors make 5 factors. That is the whole rule.

🧑‍🏫 Rule 2 · Divide: SUBTRACT. Power of a power: MULTIPLYstep 1 of 5

1
divide
$\frac{x ^{21}}{x ^{2}} = x^{19}$
21 factors over 2 factors: 2 pairs cancel, 19 are left.

🧑‍🏫 Rule 3 · Zero and negative exponentsstep 1 of 2

1
zero
$x^{0} = 1$
$\frac{x ^{3}}{x ^{3}} = x^{3 - 3} = x^{0}$ , and anything divided by itself is 1.

Trap · different bases

x^{2} \cdot y^{3}

does NOT combine. And

x^{2} + x^{3}

is not

x^{5}

— the rules are for multiplying, not adding.

🎛 Move it · root form = power form

Choose the power, the root and $x$ . The two forms always give the same number.

root form

3 x^{4}

= 16

power form

x^{\frac{4}{3}}

= 16

power p (top)4root q (bottom)3x8

Same number both ways, for every x. The POWER goes on top of the fraction, the ROOT on the bottom — “power over root”.

🧑‍🏫 Rule 4 · The n-th root of x^p is x^(p/n)step 1 of 5

1
why
$(x^{1/2})^{2} = x^{\frac{1}{2} \cdot 2} = x$
Something that squares to $x$ is $x$ , so $x^{1/2} = x$ . In general $x^{1/ n} = n x$ .

🧑‍🏫 Rule 5 · Combine roots by turning them into powersstep 1 of 4

1
📋 The question · SAT · June 2024 · Math 800 Book Q7
$5 119 n (6 119 n)^{2}$
For what value of $x$ is the given expression equivalent to $(119 n)^{30 x}$ , where $n > 1$ ?
(type the answer)
the question
$5 119 n (6 119 n)^{2} = (119 n)^{30 x}$
June 2024: find $x$ .

Desmos move · check a rewrite

Graph y = (5th root of x^3) and y = x^(3/5): one curve means equivalent. Desmos accepts x^(3/5) directly.

🧑‍🏫 Rule 6 · Write everything as ONE base, then compare exponentsstep 1 of 5

1
📋 The question · SAT · May US 2024 · Math 800 Book Q6
If $n$ and $k$ are greater than 1 and $3 n^{4}$ is equivalent to $5 k^{2}$ , for what value of $a$ is $n^{2 a + 1}$ equal to $k$ ?
(type the answer)
the question
$3 n^{4} = 5 k^{2}$
May US 2024: for what $a$ is $n^{2 a + 1} = k$ ?

🧑‍🏫 Rule 7 · Same idea with one unknown rootstep 1 of 3

1
📋 The question · SAT · December 2024 · Math 800 Book Q13
In $5 x^{m}$ , $m$ is a constant and $x > 1$ . It can be rewritten as $7 x$ . What is $m$ ?
(type the answer)
the question
$5 x^{m} = 7 x$
December 2024: find $m$ .

Real question 1SAT · June 2024 · Math 800 Book Q8type the answer

5 p^{2} = t^{3/4}

In the given equation, $p > 1$ and $t > 1$ . If $t = p^{2 n - 1}$ where $n$ is a constant, what is the value of $n$ ?

🧑‍🏫 Rule 8 · Undo a square root by squaringstep 1 of 4

1
📋 The question · SAT · October US 2024 · Math 800 Book Q11
$r = 5 w - 3$
$w > \frac{3}{5}$ . Which equation correctly expresses $w$ in terms of $r$ ?
1. A) $w = - 3 r + 5$
2. B) $w = 5 r - 3$
3. C) $w = \frac{r ^{2}}{5} + \frac{3}{5}$
4. D) $w = \frac{r ^{2}}{5} + 3$
the question
$r = 5 w - 3$
October US 2024: express $w$ in terms of $r$ .

🧑‍🏫 Rule 9 · Squaring can create fake answers — always checkstep 1 of 4

1
the question
$(x - 2)^{2} = 3 x + 34$
Practice Test 9: the smallest solution?

🧑‍🏫 Rule · Equations with powers: make the bases equalstep 1 of 3

1
📋 The question · Gen Z Study
If $9^{x + 1} = 2 7^{x}$ , what is the value of $x$ ?
(type the answer)
one base
$(3^{2})^{x + 1} = (3^{3})^{x}$
9 and 27 are both powers of 3.

🧑‍🏫 Rule · A power of a product or quotientstep 1 of 3

1
product
$(ab)^{n} = a^{n} b^{n}$
Every factor inside gets the power.

Take with you

▸Multiply → add exponents; divide → subtract; power of a power → multiply.
▸ $x^{0} = 1$ ; $x^{- n} = \frac{1}{x ^{n}}$ .
▸ $n x^{p} = x^{p / n}$ — power over root.
▸Turn every root into a power, get ONE base on each side, then set the exponents equal.
▸Undo a root by squaring (or cubing), then CHECK the answers.

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