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MathProblem-Solving and Data AnalysisHard

A store raises the price of a jacket by 20%. Later, it lowers the new price by 20%. The final price is what percent of the original price?

A80%
B96%Correct
C100%The trap
D104%

Check with $100: $100then$120then$96

  1. Turn each change into a multiplier.

    Up 20% means × 1.20. Down 20% means × 0.80.

  2. Apply them in order.

    1.20 × 0.80 = 0.96

  3. Read the answer.

    The final price is 96% of the original. That’s B.

  4. Why not 100%?

    The 20% cut is taken from the higher price, so it removes more than the 20% increase added.

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98 questions across four modules, in the digital SAT’s order: Reading & Writing Module 1 and Module 2 (27 questions each), a 10-minute break, then Math Module 1 and Module 2 (22 questions each). On official timing that’s 2 hr 14 min of testing. You get estimated section scores and a total at the end.

Yes. In the standard full-length tests, Module 2 of each section is the harder version if you get 60% or more of Module 1 right, and the easier version if you don’t. Challenge tests use the hardest material throughout.

Official timing, extended 1.5× time, or untimed, on every test, including the free ones. Extended time allows 48 minutes for each Reading & Writing module and about 52 for each Math module. Untimed removes the clock so you can focus on accuracy.

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Equivalent expressions · Lesson 5

Exponents and radicals

When the base is the same, you can rewrite products, quotients and powers by working with the exponents. Multiplying powers adds exponents. Dividing subtracts them. Raising a power to a power multiplies them. Every factor inside parentheses gets the outside exponent, numbers included: (2x3)4=16x12(2x^3)^4 = 16x^{12}.

A negative exponent means a reciprocal, not a negative number: x−2=1x2x^{-2} = \frac{1}{x^2}. A zero exponent gives 1: x0=1x^0 = 1 for x≠0x \ne 0.

A fractional exponent is a root. In xmnx^{\frac{m}{n}}, the denominator nn is the root and the numerator mm is the power: x23=x23x^{\frac{2}{3}} = \sqrt[3]{x^2}, and x=x12\sqrt{x} = x^{\frac{1}{2}}. Turn every root into a fractional exponent, then use the same rules.

To simplify a square root, pull out perfect-square factors: 72=36⋅2=62\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}, and for x>0x > 0, x5=x4⋅x=x2x\sqrt{x^5} = \sqrt{x^4 \cdot x} = x^2\sqrt{x}. Roots split across multiplication, ab=ab\sqrt{ab} = \sqrt{a}\sqrt{b} for a,b≥0a, b \ge 0, but not across addition: 9+16=5\sqrt{9 + 16} = 5, not 3+43 + 4.

Questions on this skill usually say x>0x > 0 so that every root is defined. Keep that condition in mind when you check with numbers.

Exponent rules (for x>0x > 0)
RuleExample
xm⋅xn=xm+nx^m \cdot x^n = x^{m + n}x4⋅x3=x7x^4 \cdot x^3 = x^7
xmxn=xm−n\frac{x^m}{x^n} = x^{m - n}x8x3=x5\frac{x^8}{x^3} = x^5
(xm)n=xmn(x^m)^n = x^{mn}(x4)3=x12(x^4)^3 = x^{12}
(ax)n=anxn(ax)^n = a^n x^n(5x)2=25x2(5x)^2 = 25x^2
x−n=1xnx^{-n} = \frac{1}{x^n}x−2=1x2x^{-2} = \frac{1}{x^2}
xmn=xmnx^{\frac{m}{n}} = \sqrt[n]{x^m}x34=x34x^{\frac{3}{4}} = \sqrt[4]{x^3}

Worked example Easy

Which expression is equivalent to (3x4)2⋅x3(3x^4)^2 \cdot x^3?

  1. 3x113x^{11}
  2. 6x116x^{11}
  3. 9x99x^{9}
  4. 9x119x^{11}Answer

How to solve it

  1. Raise each factor in the parentheses to the second power: 32=93^2 = 9 and (x4)2=x8(x^4)^2 = x^8, so (3x4)2=9x8(3x^4)^2 = 9x^8.
  2. Multiply by x3x^3 by adding exponents: 9x8+3=9x119x^{8 + 3} = 9x^{11}.

Why each choice is right or wrong

  • A. Incorrect. This squares x4x^4 but not the 3. The exponent applies to every factor inside the parentheses: 32=93^2 = 9.
  • B. Incorrect. This multiplies 3 by 2 instead of squaring it. 32=93^2 = 9, not 6.
  • C. Incorrect. This adds the exponents in (x4)2(x^4)^2 to get x6x^6, then x6⋅x3=x9x^6 \cdot x^3 = x^9. A power of a power multiplies the exponents: (x4)2=x8(x^4)^2 = x^8.
  • D. Correct. (3x4)2=9x8(3x^4)^2 = 9x^8, and 9x8⋅x3=9x119x^8 \cdot x^3 = 9x^{11}.

Worked example Medium

For x>0x > 0, which expression is equivalent to (x3)2⋅x2\left(\sqrt[3]{x}\right)^2 \cdot x^2?

  1. x43x^{\frac{4}{3}}
  2. x83x^{\frac{8}{3}}Answer
  3. x38x^{\frac{3}{8}}
  4. x73x^{\frac{7}{3}}

How to solve it

  1. Write the root as an exponent: x3=x13\sqrt[3]{x} = x^{\frac{1}{3}}.
  2. Apply the outer square by multiplying exponents: (x13)2=x23\left(x^{\frac{1}{3}}\right)^2 = x^{\frac{2}{3}}.
  3. Multiply by x2x^2 by adding exponents: 23+2=83\frac{2}{3} + 2 = \frac{8}{3}, so the result is x83x^{\frac{8}{3}}.

Why each choice is right or wrong

  • A. Incorrect. This multiplies 23\frac{2}{3} by 2 instead of adding. When you multiply x23x^{\frac{2}{3}} by x2x^2, the exponents add: 23+2=83\frac{2}{3} + 2 = \frac{8}{3}.
  • B. Correct. (x3)2=x23\left(\sqrt[3]{x}\right)^2 = x^{\frac{2}{3}}, and x23⋅x2=x23+2=x83x^{\frac{2}{3}} \cdot x^2 = x^{\frac{2}{3} + 2} = x^{\frac{8}{3}}.
  • C. Incorrect. This is the reciprocal of the right exponent. In a fractional exponent the root goes in the denominator, so a cube root gives thirds: 83\frac{8}{3}, not 38\frac{3}{8}.
  • D. Incorrect. This drops the outer square and uses x13⋅x2=x73x^{\frac{1}{3}} \cdot x^2 = x^{\frac{7}{3}}. The square makes the first factor x23x^{\frac{2}{3}}.

Worked example Hard

xa2xb2=x30\frac{x^{a^2}}{x^{b^2}} = x^{30}

In the given equation, x>1x > 1, and aa and bb are positive constants with a+b=10a + b = 10. What is the value of a−ba - b?

  1. 33Answer
  2. 30\sqrt{30}
  3. 2020
  4. 3030

How to solve it

  1. Dividing powers of the same base subtracts exponents, so xa2−b2=x30x^{a^2 - b^2} = x^{30}. Because x>1x > 1, equal powers of xx need equal exponents: a2−b2=30a^2 - b^2 = 30.
  2. Factor the difference of squares: (a−b)(a+b)=30(a - b)(a + b) = 30.
  3. Substitute a+b=10a + b = 10: 10(a−b)=3010(a - b) = 30, so a−b=3a - b = 3.

Why each choice is right or wrong

  • A. Correct. The exponents give a2−b2=30a^2 - b^2 = 30, which factors as (a−b)(a+b)=30(a - b)(a + b) = 30. With a+b=10a + b = 10, a−b=3a - b = 3. Check: a=6.5a = 6.5 and b=3.5b = 3.5 give 42.25−12.25=3042.25 - 12.25 = 30.
  • B. Incorrect. This writes a2−b2a^2 - b^2 as (a−b)2(a - b)^2, so (a−b)2=30(a - b)^2 = 30 and a−b=30a - b = \sqrt{30}. But a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b).
  • C. Incorrect. This subtracts a+ba + b from a2−b2a^2 - b^2: 30−10=2030 - 10 = 20. The two factors are multiplied, (a−b)(a+b)=30(a - b)(a + b) = 30, so divide instead.
  • D. Incorrect. 30 is a2−b2a^2 - b^2, the difference of the exponents. The question asks for a−ba - b.

Trap Adding exponents when you should multiply

x4⋅x3=x7x^4 \cdot x^3 = x^7 adds the exponents, but (x4)3=x12(x^4)^3 = x^{12} multiplies them. The SAT often includes the answer from using the wrong rule. Ask yourself: am I multiplying two powers, or raising a power to a power?

Trap Forgetting the coefficient or flipping the fraction

(3x4)2(3x^4)^2 squares the 3 as well: 9x89x^8, not 3x83x^8 or 6x86x^8. And x34=x34\sqrt[4]{x^3} = x^{\frac{3}{4}}: the root index goes in the denominator. x43x^{\frac{4}{3}} is the cube root of x4x^4, a different expression.

Desmos When Desmos is faster

Graph the original and each choice as y=y = lines. Type a square root with sqrt, and type any other root as a fractional exponent, such as x^(2/3), with parentheses around the exponent.

Look only at x>0x > 0, the region the question covers. The equivalent choice lies on top of the original there.

Desmos can't settle questions whose exponents are letters, such as xa2x^{a^2}. Desmos treats an undefined letter as a constant and offers a slider for it, which doesn't solve for it. Work those with the rules.

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