SAT Math formula sheet · all 24 · Advanced Math · #6 of 24

Converting radicals to exponents on the SAT: How It Is Tested

Equivalent expressions

Exponential to Radical diagram for the SAT: Never leave a radical as a radical. Root on the outside, power on the inside, and as an exponent you write power divided
What it is really testingNever leave a radical as a radical. Root on the outside, power on the inside, and as an exponent you write power divided by root. Also remember that a square root has a root of 2 even though nobody writes a 2 on the outside.

The question from the sheet

Hard
SAT Advanced Mathanswer 11/6
If x7x53=xab\frac{\sqrt{x^{7}}}{\sqrt[3]{x^{5}}} = x^{\frac{a}{b}} for all positive values of xx, where ab\frac{a}{b} is in lowest terms, what is the value of ab\frac{a}{b}?
Student-produced response: 11/6

How the diagram applies

  1. Radical to exponent, power over root: x7=x7/2\sqrt{x^{7}} = x^{7/2} and x53=x5/3\sqrt[3]{x^{5}} = x^{5/3}.
  2. Dividing like bases: subtract the exponents. 72−53=216−106=116\tfrac{7}{2} - \tfrac{5}{3} = \tfrac{21}{6} - \tfrac{10}{6} = \tfrac{11}{6}.
  3. ab=116\tfrac{a}{b} = \tfrac{11}{6}.

Answer: 11/6

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Common Questions

It comes up as an SAT equivalent expressions question. Never leave a radical as a radical. Root on the outside, power on the inside, and as an exponent you write power divided by root. Also remember that a square root has a root of 2 even though nobody writes a 2 on the outside.

Radical to exponent, power over root: x^7 = x^7/2 and [3]x^5 = x^5/3. Dividing like bases: subtract the exponents. 72 - 53 = 216 - 106 = 116. ab = 116.

No. The SAT reference sheet is twelve geometry formulas and three notes, and Exponential to Radical is not one of them. It is one of the 24 things the test expects you to know without being given them. See what the reference sheet does cover at /blog/sat-math-reference-sheet.