Exponents and Roots MCQs with Answers & Explanations (Page 5)

Simplify both terms first: sqrt(18) = 3*sqrt(2) and sqrt(8) = 2*sqrt(2). Subtracting them gives 3*sqrt(2) - 2*sqrt(2) = 1*sqrt(2), which is just sqrt(2).

When multiplying square roots, multiply the numbers inside the radicals together. This gives sqrt(36), and since 36 is a perfect square, the square root is 6.

Multiply the numbers inside the roots to get sqrt(100). The principal square root of 100 is 10, completing the simplification.

Use the multiplication property of radicals to combine them into one root: sqrt(2 * 8) = sqrt(16). The square root of 16 perfectly evaluates to 4.

Multiplying a square root by itself cancels out the radical, effectively squaring the square root. sqrt(6) * sqrt(6) is sqrt(36), which immediately simplifies to 6.

For division of radicals, divide the radicands (the numbers inside) under a single root. This makes it sqrt(24 / 6) = sqrt(4). The square root of 4 is exactly 2.

Using the quotient property, merge the terms into one square root: sqrt(50 / 2). This simplifies to sqrt(25), which has a value of 5.

To remove the radical from the denominator, multiply both numerator and denominator by sqrt(2). This results in (1 * sqrt(2)) / (sqrt(2) * sqrt(2)), which simplifies to sqrt(2) / 2.

Multiply numerator and denominator by sqrt(3) to rationalize. This gives (3*sqrt(3)) / 3. The 3s cancel out, leaving just sqrt(3).

Multiply both the top and the bottom by sqrt(5). This produces (5*sqrt(5)) / 5. Dividing by 5 leaves only the term sqrt(5).

Multiply numerator and denominator by sqrt(7) to rationalize. The numerator becomes 2*sqrt(7) and the denominator becomes 7, creating the fraction 2*sqrt(7) / 7.

To solve, express both sides with the same base. Since 16 is equal to 2^4, the equation becomes 2^x = 2^4. Thus, the exponents must be equal, making x = 4.

Express 27 as a power of 3, which is 3^3. Setting the exponents equal from the equation 3^x = 3^3 tells us that x equals 3.

Write 125 with a base of 5. Because 5 * 5 * 5 = 125, we know 125 is 5^3. Equating the exponents from 5^x = 5^3 gives x = 3.

Count the number of zeroes in 10000 to find its power of 10. There are four zeroes, so 10000 is 10^4. Therefore, from 10^x = 10^4, x must be 4.

Convert 64 into a power of 4. We know that 4 * 4 * 4 equals 64, making it 4^3. Setting the exponents equal, we determine that x = 3.

To isolate x, perform the inverse operation of a square root by squaring both sides of the equation. Squaring 5 gives 25, so x = 25.

To remove the cube root, cube both sides of the equation. The cube of 3 is 3 * 3 * 3, which equals 27. Therefore, x = 27.

Square both sides to eliminate the radical, creating the equation x - 2 = 16. Then, add 2 to both sides to solve for x, resulting in x = 18.

First, divide both sides by 2 to isolate the radical, giving sqrt(x) = 5. Next, square both sides to find x. Squaring 5 results in x = 25.