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Advanced Math · Exponential And Radical Equations

SAT Exponential And Radical Equations Practice Questions (Free + Explanations) | Quiz 9

Question 12345 of 5

Question 1 of 5

For all real numbers such that the expression is defined, the expression

is equivalent to which of the following?

Explanation

Let . Since for every real , rewrite the expression as

Now factor the quadratic inside the radical:

So the original expression becomes

Because , we have , so . Substituting back gives

Concept summary

To rewrite a radical expression in equivalent form, look for a perfect-square trinomial. After factoring, use domain information to decide whether an absolute value can be removed.

Question 2 of 5

If , what is the value of ?

Explanation

Let and . Then , and the given equation becomes . We want . Use the identity

Substituting the known values gives

So,

Because and , the value of is .

Concept summary

When an expression involves , rewrite it with a substitution and use identities such as , especially when the product is easy to find.

Question 3 of 5

For and , which expression is equivalent to

Explanation

Factor the denominator using the difference of squares:

Then

Since , the factor is not , so it can be canceled. This gives

Concept summary

When expressions involve radicals, rewrite powers like as when helpful. Then use factoring, especially the difference of squares, to simplify equivalent forms.

Question 4 of 5

If , what is the value of ?

Explanation

To undo the square root, square both sides of the equation:

This gives .

Concept summary

When a variable is inside a square root, square both sides to isolate the variable.

Question 5 of 5

A student records values of the function in the table below.

Based on the pattern in the table and the function, what value should replace the question mark?

Explanation

Use the function rule with :

This also fits the pattern shown in the table, since for , , and for , . So the missing value is .

Concept summary

For an exponential expression inside a radical, evaluate the exponent first, then simplify the quantity under the square root.

Your results

0of 5 correct

Estimated SAT Math band

500-550

Illustrative range from this short quiz—not an official College Board score.

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Your results

1of 5 correct

Estimated SAT Math band

500-550

Illustrative range from this short quiz—not an official College Board score.

Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.

More SAT Math practice

Your results

2of 5 correct

Estimated SAT Math band

600-650

Illustrative range from this short quiz—not an official College Board score.

Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.

More SAT Math practice

Your results

3of 5 correct

Estimated SAT Math band

600-650

Illustrative range from this short quiz—not an official College Board score.

Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.

More SAT Math practice

Your results

4of 5 correct

Estimated SAT Math band

700+

Illustrative range from this short quiz—not an official College Board score.

Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.

More SAT Math practice

Your results

5of 5 correct

Estimated SAT Math band

700+

Illustrative range from this short quiz—not an official College Board score.

Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.

More SAT Math practice