Video: Laws of Exponents

Complete the following: −64𝑏⁻⁶ = (_)³.

03:41

Video Transcript

Complete the following. Negative 64𝑏 to the power of negative six is equal to blank cubed.

In order to answer this question, we need to recall the laws of exponents or indices. Firstly, 𝑥 to the power of 𝑛 multiplied by 𝑥 to the power of 𝑚 is equal to 𝑥 to the power of 𝑛 plus 𝑚. We add the exponents. Also, 𝑥 to the power of 𝑛 raised to the power of 𝑚 is equal to 𝑥 to the power of 𝑛𝑚. We multiply the exponents.

One way to solve the problem would be to consider the fact that cubing means multiplying something by itself and itself again. What number cubed gives us negative 64? Four cubed is equal to 64, as four multiplied by four multiplied by four equals 64. This means that negative four cubed is equal to negative 64. Negative four multiplied by negative four is equal to positive 16. Multiplying this by negative four is negative 64.

We now need to consider the 𝑏 to the power of negative six part and the first law of exponents. Negative two plus negative two plus negative two is equal to negative six. This means that 𝑏 to the power of negative two multiplied by 𝑏 to the power of negative two multiplied by 𝑏 to the power of negative two gives us 𝑏 to the power of negative six. This means that negative four 𝑏 to the power of negative two cubed is equal to negative 64𝑏 to the power of negative six.

We can check this by using the second law that we wrote down. Multiplying the powers, negative two and three, gives us negative six. Therefore, this answer is correct. The missing term that would make the equation correct is negative four 𝑏 to the power of negative two. An alternative method would be to solve for 𝑥 the equation negative 64 𝑏 to the power of negative six equals 𝑥 cubed. The opposite of cubing is cube rooting, so we would need to cube root both sides of the equation. On the right-hand side, the cube root of 𝑥 cubed is equal to 𝑥.

On the left-hand side, we need to work out the cube root of negative 64 and the cube root of 𝑏 to the power of negative six. The cube root of negative 64 is negative four, as negative four cubed equals negative 64. The cube root of 𝑥 is the same as 𝑥 to the power of one-third. This means that the cube root of 𝑏 to the power of negative six is 𝑏 to the power of negative six to the power of a third. To simplify this, we multiply the powers. We multiply one-third by negative six. This is equal to negative two. We have therefore solved the equation for 𝑥 such that 𝑥 is equal to negative four 𝑏 to the power of negative two. Which was the correct answer.

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