Digital SAT Maths -Unit 2 - 2.2 Exponents and Radicals- Study Notes- New Syllabus
Digital SAT Maths -Unit 2 – 2.2 Exponents and Radicals- Study Notes- New syllabus
Digital SAT Maths -Unit 2 – 2.2 Exponents and Radicals- Study Notes – per latest Syllabus.
Key Concepts:
Laws of exponents
Negative & fractional exponents
Simplifying radicals
Laws of Exponents
Exponents represent repeated multiplication.
\( a^3 = a \cdot a \cdot a \)
The DIGITAL SAT frequently tests whether you know how to rewrite and simplify expressions using exponent rules.
1. Product Rule
\( a^m \cdot a^n = a^{m+n} \)
(Add the exponents when bases are the same)
2. Quotient Rule
\( \dfrac{a^m}{a^n} = a^{m-n} \)
3. Power of a Power
\( (a^m)^n = a^{mn} \)
4. Power of a Product
\( (ab)^n = a^n b^n \)
5. Zero Exponent Rule
\( a^0 = 1 \) (for \( a \ne 0 \))
Important Warning
These rules only work when the bases are the same.
\( 2^3 \cdot 3^3 \ne 6^6 \)
DIGITAL SAT Tip
Many SAT questions do not look like exponent questions. They ask you to identify an equivalent expression.
Example 1:
Simplify \( x^4 \cdot x^3 \).
▶️ Answer/Explanation
Add exponents:
\( x^{4+3}=x^7 \)
Answer: \( x^7 \)
Example 2:
Simplify \( \dfrac{y^8}{y^5} \).
▶️ Answer/Explanation
Subtract exponents:
\( y^{8-5}=y^3 \)
Answer: \( y^3 \)
Example 3:
Simplify \( (2x^2)^3 \).
▶️ Answer/Explanation
\( (2^3)(x^{2\cdot3}) \)
\( 8x^6 \)
Answer: \( 8x^6 \)
Negative & Fractional Exponents
The DIGITAL SAT often rewrites radicals as exponents and exponents as radicals. You must be comfortable moving between the two forms.
Negative Exponents
A negative exponent means reciprocal.
\( a^{-n} = \dfrac{1}{a^n} \)
Examples:
\( x^{-2} = \dfrac{1}{x^2} \)
\( \dfrac{1}{x^{-3}} = x^3 \)
Fractional Exponents
Fractional exponents represent roots.
\( a^{1/2} = \sqrt{a} \)
\( a^{1/3} = \sqrt[3]{a} \)
General rule:
\( a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m \)
Combined Form
\( a^{-m/n} = \dfrac{1}{\sqrt[n]{a^m}} \)
DIGITAL SAT Tip
If the answer choices contain radicals, rewrite exponents as roots. If the answers contain exponents, rewrite radicals as powers.
Example 1:
Rewrite \( x^{-3} \) without negative exponents.
▶️ Answer/Explanation
\( x^{-3}=\dfrac{1}{x^3} \)
Answer: \( \dfrac{1}{x^3} \)
Example 2:
Rewrite \( \sqrt{x^5} \) using exponents.
▶️ Answer/Explanation
\( \sqrt{x^5}=x^{5/2} \)
Answer: \( x^{5/2} \)
Example 3:
Simplify \( 16^{3/4} \).
▶️ Answer/Explanation
\( 16^{3/4}=(\sqrt[4]{16})^3 \)
\( \sqrt[4]{16}=2 \)
\( 2^3=8 \)
Answer: 8
Simplifying Radicals
A radical is a root expression such as a square root or cube root.
\( \sqrt{a} \), \( \sqrt[3]{a} \)
To simplify a radical, we remove perfect powers from inside the root.
Key Idea
If a number contains a perfect square factor:
\( \sqrt{ab}=\sqrt{a}\sqrt{b} \)
We take the square root of the perfect square.
Perfect Squares to Remember
\( 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 \)
Radicals with Variables
\( \sqrt{x^2}=x \)
\( \sqrt{x^6}=x^3 \)
Rationalizing the Denominator
We do not leave a radical in the denominator.
\( \dfrac{1}{\sqrt{a}}=\dfrac{\sqrt{a}}{a} \)
DIGITAL SAT Tip
Answer choices often differ only by simplified radical form, so full simplification matters.
Example 1:
Simplify \( \sqrt{72} \).
▶️ Answer/Explanation
\( 72=36\cdot2 \)
\( \sqrt{72}=\sqrt{36}\sqrt{2} \)
\( =6\sqrt{2} \)
Answer: \( 6\sqrt{2} \)
Example 2:
Simplify \( \sqrt{50x^2} \).
▶️ Answer/Explanation
\( 50=25\cdot2 \)
\( \sqrt{50x^2}=\sqrt{25}\sqrt{2}\sqrt{x^2} \)
\( =5x\sqrt{2} \)
Answer: \( 5x\sqrt{2} \)
Example 3 (Rationalize Denominator):
Simplify \( \dfrac{4}{\sqrt{5}} \).
▶️ Answer/Explanation
Multiply by \( \dfrac{\sqrt{5}}{\sqrt{5}} \)
\( \dfrac{4\sqrt{5}}{5} \)
Answer: \( \dfrac{4\sqrt{5}}{5} \)