Exponents and Radicals

Exponents & Radicals

Part 1 of 7 — Exponent Rules

The Core Rules

SAT Trap

$(x + y)^2 \neq x^2 + y^2$ — you MUST FOIL!

$(x + y)^2 = x^2 + 2xy + y^2$

But $(xy)^2 = x^2 y^2$ ✓ — distribution works for products, NOT sums.

Worked Example 1

Simplify $\frac{(2x^3)^2 \cdot x^4}{4x^5}$.

Worked Example 2

Rewrite $\frac{1}{x^{-3}}$ with positive exponents.

Exponent Rules 🎯

Negative Exponents in Fractions

A negative exponent "flips" a factor between numerator and denominator:

$\frac{x^{-2}}{y^{-3}} = \frac{y^3}{x^2}$

Worked Example 3

Simplify $\frac{3a^{-2}b^3}{6a^4b^{-1}}$.

Common Base Conversions

Harder Exponent Problems 🎯

Which Rule? 🔍

Identify the exponent rule used in each simplification.

Key Takeaways — Part 1

• Exponents do NOT distribute over addition: $(a + b)^n \neq a^n + b^n$
• Convert to matching bases when comparing or solving equations

Exponents & Radicals

Part 2 of 7 — Radicals and Rational Exponents

Radical ↔ Exponent Conversion

$a^{1/n} = \sqrt[n]{a} \qquad a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Examples:

• $x^{1/2} = \sqrt{x}$
• $x^{2/3} = \sqrt[3]{x^2}$
• $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$

Simplifying Radicals

$\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$

Look for perfect square factors: 4, 9, 16, 25, 36, 49, 64, 81, 100...

Rationalizing the Denominator

$\frac{3}{\sqrt{5}} = \frac{3}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5}$

Worked Example 1

Evaluate $16^{3/4}$.

Worked Example 2

Rationalize $\frac{2}{3 + \sqrt{2}}$.

Radicals & Rational Exponents 🎯

Converting Between Forms

The SAT often asks you to rewrite an expression. Know these equivalences:

Worked Example 3

Rewrite $\frac{x^2}{\sqrt[3]{x}}$ using a single exponent.

Worked Example 4

Simplify $\sqrt{x^4 y^6}$.

Conversion Practice 🎯

Radical or Exponent? 🔍

Convert each expression to the other form.

Key Takeaways — Part 2

Exponents & Radicals

Part 3 of 7 — Scientific Notation & Large/Small Numbers

Scientific Notation: $a \times 10^n$

Where $1 \leq |a| < 10$ and $n$ is an integer.

• Large: $4{,}500{,}000 = 4.5 \times 10^6$
• Small: $0.00032 = 3.2 \times 10^{-4}$

Operations with Scientific Notation

Multiply: $(3 \times 10^4)(2 \times 10^5) = 6 \times 10^9$

Divide: $\frac{8 \times 10^7}{4 \times 10^3} = 2 \times 10^4$

Powers of 10 Shortcuts

• Moving decimal right = smaller exponent
• Moving decimal left = larger exponent
• $10^3 = 1{,}000$ (3 zeros)

Worked Example 1

Multiply $(4.2 \times 10^5)(3 \times 10^{-2})$.

Worked Example 2

How many times larger is $6 \times 10^8$ than $3 \times 10^5$?

Scientific Notation 🎯

Adding/Subtracting in Scientific Notation

To add or subtract, the exponents must match first.

Worked Example 3

Add $3.5 \times 10^4 + 2.1 \times 10^3$.

Worked Example 4

A cell has mass $8.3 \times 10^{-12}$ grams. How many cells in 1 gram?

SAT Tip: When comparing very large or very small numbers, look at the exponent first — bigger exponent = bigger number (for positive coefficients).

Real-World Scientific Notation 🎯

Convert and Compare 🔍

Select the correct scientific notation or comparison.

Key Takeaways — Part 3

• Always adjust so coefficient is between 1 and 10
• SAT context: distances, populations, atomic sizes — the math stays the same

Exponents & Radicals

Part 4 of 7 — Solving Equations with Exponents

Strategy 1: Make the Bases Match

If $2^{3x} = 8^{x+1}$:

• Rewrite $8 = 2^3$: $2^{3x} = (2^3)^{x+1} = 2^{3x+3}$
• Bases match → $3x = 3x + 3$? That gives $0 = 3$, so no solution.

Strategy 2: Bracket the Answer with Whole-Number Powers

If $3^x = 15$, no whole-number exponent works, because $15$ is not a power of $3$. The SAT will not ask for an exact value here, but it might ask:

• "Between which two integers is $x$?" Since $3^2 = 9 < 15 < 27 = 3^3$, $x$ is between 2 and 3.

Strategy 3: Exponential Equations from Context

"A population doubles every 5 years. Starting at 1000, when will it reach 8000?"

$1000 \cdot 2^{t/5} = 8000$ → $2^{t/5} = 8 = 2^3$ → $t/5 = 3$ → $t = 15$ years.

Worked Example 1

Solve $25^{x-1} = 125$.

Worked Example 2

If $2^a = 5$, express $2^{3a}$ in terms of a number.

Exponential Equations 🎯

Exponential Growth and Decay

Worked Example 3

A car worth $20,000 depreciates 15% per year. When is it worth $10,000?

Worked Example 4

$2^x \cdot 4^{x+1} = 8^3$. Find $x$.

Growth & Decay 🎯

Match the Base 🔍

Rewrite each number as a power of 2 or 3.

Key Takeaways — Part 4

Exponents & Radicals

Part 5 of 7 — Radical Equations

Solving Radical Equations

1. Isolate the radical on one side
2. Square (or cube, etc.) both sides
3. Solve the resulting equation
4. CHECK for extraneous solutions!

Example: $\sqrt{x + 3} = x - 3$

Square both sides: $x + 3 = (x - 3)^2 = x^2 - 6x + 9$

Rearrange: $x^2 - 7x + 6 = 0$ → $(x - 1)(x - 6) = 0$

Check $x = 1$: $\sqrt{4} = 1 - 3 = -2$? No! $2 \neq -2$ ❌ Extraneous!

Check $x = 6$: $\sqrt{9} = 6 - 3 = 3$? Yes! ✓

Why Extraneous Solutions Appear

Squaring both sides can introduce false solutions because $(-3)^2 = 3^2 = 9$. The squaring step "loses" the sign information.

Worked Example 1

Solve $\sqrt{5x + 1} = x + 1$.

Radical Equations 🎯

Equations with Two Radicals

When there are two radicals, isolate one, square, then isolate the other and square again.

Worked Example 2

Solve $\sqrt{x + 5} - \sqrt{x} = 1$.

Cube Root Equations

No extraneous solutions with cube roots (cubing preserves sign).

Worked Example 3

Solve $\sqrt[3]{2x - 1} = 3$.

SAT Tip: The SAT will specifically design problems to test whether you check for extraneous solutions. Always substitute back into the original equation.

Trickier Radical Equations 🎯

Valid or Extraneous? 🔍

For each solution, determine if it is valid.

Key Takeaways — Part 5

• $\sqrt{x} \geq 0$ always — if the other side is negative, the solution is extraneous
• Two radicals? Isolate one, square, isolate the other, square again
• Cube roots: no extraneous solutions (cubing preserves sign)

Exponents & Radicals

Part 6 of 7 — Simplifying Complex Expressions

Combining Radicals

• $a\sqrt{n} + b\sqrt{n} = (a + b)\sqrt{n}$ (like terms!)
• $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$
• $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

Example: $3\sqrt{2} + 5\sqrt{2} - \sqrt{2} = 7\sqrt{2}$

But: $3\sqrt{2} + 5\sqrt{3}$ cannot be simplified further.

Simplify Before Combining

$\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$

Worked Example 1

Simplify $\sqrt{18} + \sqrt{50} - \sqrt{8}$.

Worked Example 2

Simplify $\left(\frac{x^4}{y^2}\right)^{3/2}$.

Simplifying Expressions 🎯

Multiplying Radical Expressions

Use FOIL when multiplying binomials with radicals.

Worked Example 3

Expand $(3 + \sqrt{5})(3 - \sqrt{5})$.

This is the conjugate pattern — the radical disappears!

Worked Example 4

Expand $(2\sqrt{3} + 1)^2$.

Worked Example 5

Simplify $\frac{x^{2/3} \cdot x^{-1/6}}{x^{1/2}}$.

Complex Simplification 🎯

Can These Be Combined? 🔍

Determine whether each pair can be simplified into a single term.

Key Takeaways — Part 6

• Convert to exponent form for complex fraction simplification
• Conjugate multiplication eliminates radicals in denominators
• $\sqrt{a} + \sqrt{b} \neq \sqrt{a + b}$ — radicals don't distribute over addition!

Exponents & Radicals

Part 7 of 7 — Review & SAT-Level Practice

Quick Reference Card

Common SAT Exponent Traps

1. $(x + y)^2 \neq x^2 + y^2$ — must FOIL
2. $(-x)^2 = x^2$ but $-x^2 = -(x^2)$ — order of operations!
3. $\sqrt{x^2} = |x|$, not just $x$
4. $a^0 = 1$ for ALL nonzero $a$, including negatives: $(-5)^0 = 1$

Worked Example 1 — Trap-Style Problem

If $(-3)^4 - 3^4 + (-3)^3 + 3^3 = ?$

Shortcut: Even powers make signs agree; odd powers make signs cancel.

Worked Example 2 — Multi-Rule Problem

If $\frac{(2x^3)^2 \cdot x^{-4}}{4x^2} = x^n$, find $n$.

Mixed Review 🎯

SAT Strategy: Rewrite Everything as Powers of Small Primes

Many SAT problems look complex but simplify once you rewrite bases as powers of 2, 3, or 5.

Worked Example 3

If $8^x = 32$, what is $x$?

Worked Example 4

Simplify $\frac{9^{n+1}}{3^{2n-1}}$.

Worked Example 5

If $\sqrt{x^2 + 6x + 9} = 7$, find $x$.

SAT-Level Challenge 🎯

Which Strategy? 🔍

For each problem, choose the best first step.

Key Takeaways — Full Topic Review

SAT Strategies:

• Rewrite all bases as powers of small primes (2, 3, 5)
• Convert between radical and exponent form freely
• Check extraneous solutions after squaring
• For $(-1)^n$: odd → $-1$, even → $1$