Video: Simplifying Numerical Expressions Involving Square and Cube Roots

Express ∛(875) − (64/√(8)) + √(512) + ∛(448) in its simplest form.

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Video Transcript

Express the cube root of 875 minus 64 divided by root eight plus the square root of 512 plus the cube root of 448 in its simplest form.

In order to answer this question, we need to simplify each of the terms individually. The cube root of 875 can be rewritten as the cube root of 125 multiplied by the cube root of seven. This is because 125 multiplied by seven is equal to 875. 125 is a cube number as five cubed equals 125. Therefore, the cube root of 125 is equal to five. The cube root of 875 in its simplest form is therefore five multiplied by the cube root of seven.

In order to simplify the second term, 64 divided by root eight, we need to rationalize the denominator. We do this by multiplying the top and the bottom of the fraction by root eight. 64 multiplied by root eight is equal to 64 root eight. And root eight multiplied by root eight is equal to eight. We can then divide 64 by eight leaving us with eight root eight. Root eight can also be written as root four multiplied by root two. The square root of four is equal to two. Therefore, root eight is equal to two root two. Multiplying this by eight tells us that 64 divided by root eight in its simplest form is 16 root two.

The third term, the square root of 512, can be rewritten as the square root of 256 multiplied by the square root of two. 256 is a square number as 16 multiplied by 16 equals 256. Therefore, the square root of 256 is 16. This means that the square root of 512 in its simplest form is 16 root two.

Finally, we need to simplify the cube root of 448. 64 multiplied by seven is equal to 448. Therefore, the cube root of 448 can be rewritten as the cube root of 64 multiplied by the cube root of seven. 64 is a cube number as four cubed equals 64. Therefore, the cube root of 64 is equal to four. We can therefore write the cube root of 448 in its simplest form as four multiplied by the cube root of seven.

We now have simplified expressions for all four terms. The cube root of 875 minus 64 divided by root eight plus the square root of 512 plus the cube root of 448 is therefore equal to five multiplied by the cube root of seven minus 16 root two plus 16 root two plus four multiplied by the cube root of seven. The two 16 root twos cancel as negative 16 root two plus 16 root two is equal to zero. Five multiplied by the cube root of seven plus four multiplied by the cube root of seven is equal to nine multiplied by the cube root of seven.

This is the answer to our expression in its simplest form, nine multiplied by the cube root of seven.

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