Exponents & Radicals - Chapter

Sep 14, 2026

Uploaded by Franzy

A focused SAT math guide explains exponent rules and radical (root) notation, including how bases and exponents interact in products, quotients, and powers. It clarifies negative-exponent behavior using order of operations (PEMDAS), then connects radicals to fractional exponents to show how roots and exponents undo each other. Worked examples demonstrate simplifying expressions into equivalent forms and extracting factors from square roots and cube roots without calculators.

LanguageEnglish

CategoryExam

FormatPDF

Pages11

Reading Time28 min

AccessFree to download

Exponents and exponent laws

Negative exponents and order of operations (PEMDAS)

Worked examples: equivalent expressions

Converting powers to common bases

Roots and radical notation

Fractional exponents connection

Simplifying square roots and cube roots

Extracting perfect factors

Practice exercises

Exponents & Radicals Given the expression xn, x is called the base and n is called the exponent or power. Here are the laws of exponents you should know: 1 | Law | Example | | --- | --- | | x1 = x | 31 = 3 | | x0 = 1 | 30 = 1 | | m · n m+nx x = x | 34 · 35 = 39 | | mx
m−n n = x
x | 373~~3~~ = 34 | | !xm"n = xmn | !32"4 = 38 | | (xy)m = xmym | (2 · 3)3 = 23 · 33 | | Å xyãm = xymm | Å 2~~3~~ã3 = 233~~3~~ | | –m 1
x = mx | 3–4 1
= 34 | The difference between (−3)2 and − 32 comes down to order of operations (PEMDAS), which dictates that parentheses are prioritized ﬁrst. So, (−3)2 = (−3) · (−3) = 9 Without parentheses, exponents take priority: −32 = −3 · 3 = −9 Notice that the negative is not applied until the exponent operation is carried through. Sometimes, the result turns out to be the same, as in: (−2)3 and −23 Both yield −8. EXAMPLE 1: Which of the following is equivalent to −x2 (−x)5 ? A) −x10 B) −x7 C) x7 D) x10 The −x2 term cannot be simpliﬁed any further because the negative is not inside any parentheses. The (−x)5 term has an odd exponent. An odd exponent yields an odd number of negative signs, which in turn yields a negative result. So,(−x)5 simpliﬁes to −x5. The entire expression then boils down to −x2 (−x)5 = −x2 · −x5 = x2+5 = x7 Answer (C) . EXAMPLE 2: Which expression is equivalent to 4x+1 · 32x, where x > 0 ? A) 4(12)x B) 4(24)x C) 4(36)x D) 8(12)x We can use our laws of exponents to simplify the expression. First, the law xm+n = xm · xn allows us to simplify 4x+1 to 4x · 41. Then the law xmn = !xm"n allows us to turn 32x into !32"x = 9x . After applying these laws, we get 4x+1 · 32x = 4x · 41 · 9x = 4 · (4x · 9x ) = 4(36)x Answer (C) . In the last step, we used the law xmym = (xy)m to get 4x · 9x = 36x . Roots are the opposites of exponents. Whereas 52 will give you 25, the square root of 25, denoted by √25, will give you back the 5 . Similarly, taking the cube root of 63 gives \#3 63 = 6 and taking the fourth root of b4 gives \#4 b4 = b. Notice how the operations cancel each other out. Instead of using the radical sign, √n ~~ ~~ , we can also denote the nth-root using the equivalent fractional 1 exponent . For example, n 1 √x = x 2 1 √3 x = x 3 2 What about x 3 ? The 2 on top means to square x. The 3 on the bottom means to cube root it: \#3 x2 We can see this more clearly if we break it down using our laws of exponents: x ~~23~~ = Äx2 ä~~13~~ = \#3 x2 The order in which we do the squaring and the cube-rooting doesn’t matter. We could’ve broken it down this way as well: x ~~23~~ = !x~~13~~ "2 = !√3 x"2 You’ll see \#3 x2 more often than!√3 x"2 because the outside cube root avoids the need for parentheses. EXAMPLE 3: Which of the following is equivalent to \#4 x5 ? A) x B) x5 − x4 C) x ~~54~~ D) x ~~45~~ The fourth root equates to a fractional exponent of 1~~4~~, so \#4 x5 = Äx5 ä~~14~~ = x ~~54~~ Answer (C) . EXAMPLE 4: The expression Ä2~~5~~x is equivalent to 5kx, where k is a constant. What is the value of k ? Because most of the laws of exponents involve terms with the same base, an often-used tactic is to convert each term to have the same base. Here, we’re asked to simplify to 5kx, so we should convert everything to have the same base of 5 . As is common in this type of question, we see that we have a perfect square, 25, which can be written as 52 . In the denominator, we have √5, which can be written as 5 ~~12~~ . 252x = !521"2x 54x 54x − ~~12~~ x 5~~72~~ x Ä√ 5 äx !52 "x = 512 x = = 7 Therefore, k = ~~ ~~ 2 . EXAMPLE 5: If the square of a is equal to the cube of b, where a ≥ 0 and b ≥ 0, for what value of x is √ax equal to b ? Essentially, we need to solve √ax = b for x given that a2 = b3. Let’s ﬁrst cube root both sides of a2 = b3 to isolate b: \#3 a2 = \#3 b3 2 a 3 = b 2 Substituting a 3 for b in the equation above, we get 2 √ax = a 3 1 2 a 2 x = a 3 Since both

FAQs

How do exponent rules handle multiplication and division of powers?

Use xm+n = xm · xn for multiplying same bases, and use xmn = (xm)n for raising a power to another power. For division with the same base, subtract exponents (implied by the rules used throughout the guide).

Why is (−3)2 different from −3^2?

Parentheses are evaluated first due to PEMDAS. (−3)2 multiplies (−3)·(−3) for a positive result, while −3^2 is interpreted as −(3·3), applying the negative after the exponent operation.

What is the relationship between radicals and fractional exponents?

The nth root of x can be written as x^(1/n). This lets you convert between radical form and fractional exponent form to simplify expressions using exponent laws.

Sep 14, 2026

Uploaded by Franzy