Lesson Video: Simplifying Numerical Expressions - Negative and Fractional Exponents Mathematics

In this video, we will learn how to use the rules of negative and fractional exponents to simplify numerical expressions.

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Video Transcript

In this video, we will learn how to use the rules of negative and fractional exponents or indices to simplify numerical expressions. We will begin by looking at the rules we can use when dealing with negative exponents and fractional exponents.

When dealing with negative exponents, we use the following rule: 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. Let’s consider a couple of examples: firstly, five to the power of negative two and then two to the power of negative four. Using our rule, five to the power of negative two can be rewritten as one over five to the power of two or five squared. As five squared is equal to 25, five to the power of negative two is one over 25 or one twenty-fifth. In the same way, two to the power of negative four is equal to one over two to the fourth power or two to the power of four. This is the same as two multiplied by two multiplied by two multiplied by two. Two to the power of four is equal to 16; therefore, two to the power of negative four is one over 16.

Let’s now consider what happens when we want to raise a fraction to a negative power. We know that one over a number is known as the reciprocal of that number. When finding the reciprocal of a fraction, we swap the numerator and denominator. This means that 𝑎 over 𝑏 to the power of negative 𝑛 is equal to 𝑏 over 𝑎 to the power of 𝑛. In our example, four over three to the power of negative two is the same as three over four to the power of two all squared. When raising any fraction to a power, we can raise the numerator and denominator to this power separately. Three-quarters or three over four squared is the same as three squared over four squared. This is equal to nine over 16 or nine sixteenths.

We will use these two rules when dealing with negative exponents of integers and fractions. When dealing with fractional exponents, we use the rule that 𝑥 to the power of one over 𝑛 is equal to the 𝑛th root of 𝑥. This means that eight to the power of one-third is the same as the cube root of eight. As two cubed is equal to eight, the cube root of eight is equal to two. This means that eight to the power of one-third equals two. When trying to calculate 𝑥 to the power of 𝑚 over 𝑛, we begin by finding the 𝑛th root of 𝑥 and then raise this answer to the power of 𝑚.

Let’s consider 16 to the power of three over two. The denominator is two, so this is the root. We need to take the square root of 16. As the numerator is three, we will then cube this answer. The square root of 16 is four and four cubed is equal to 64. Therefore, 16 to the power of three over two is equal to 64.

We could also do these two operations in the opposite order. We could raise 𝑥 to the power of 𝑚 and then take the 𝑛th root. In our example, we would calculate 16 cubed and then square root the answer. Whilst this gives us the same answer, it is far more complicated to cube 16 and then square root 4096 than square root 16 and then cube four. In the vast majority of questions, it will be easier to find the root first.

Let’s now consider what happens when we have a negative fractional exponent, for example, 32 to the power of negative two-fifths. We recall that when we had a negative exponent, we looked at the reciprocal. 32 to the power of negative two-fifths is the same as one over 32 to the power of two-fifths. Our next step is to find the fifth root of 32 and then square the answer. The fifth root of 32 is equal to two. So, we are left with one over two squared. 32 to the power of negative two-fifths is, therefore, equal to one-quarter. 𝑥 to the power of negative 𝑚 over 𝑛 is equal to one over the 𝑛th root of 𝑥 raised to the power of 𝑚. We can use these three rules when solving problems with fractional exponents.

We will now look at some questions involving negative and fractional exponents.

Which of the following is equal to four multiplied by the square root of six to the power of negative one? Is it (A) 24, (B) four root six, (C) two root six over three, or (D) two-thirds?

When dealing with negative exponents, we use the rule that 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. This means that 𝑥 to the power of negative one is equal to one over 𝑥 to the power of one which is the same as one over 𝑥. The square root of six to the power of negative one is equal to one over the square root of six. This means that when we multiply this by four, we get four over the square root of six or four over root six. As we have a radical or surd on the denominator, we need to rationalize this fraction. We do so by multiplying the numerator and denominator by the square root of six.

This gives us four root six over six as multiplying the square root of six by the square root of six is six. This comes from the fact that one of our laws of indices states that the square root of 𝑥 multiplied by the square root of 𝑥 is equal to 𝑥. We can simplify this fraction by dividing the numerator and denominated by two. This leaves us with two root six over three. The correct answer is, therefore, option (C). Four multiplied by the square root of six raised to the power of negative one is equal to two root six over three.

Our next question involves a fractional power that is negative.

Evaluate 64 over 27 to the power of negative two-thirds.

In order to answer this question, we need to use our knowledge of fractional and negative exponents or indices. Firstly, when we raise a fraction to a negative power, this is the same as the reciprocal of the fraction to the positive of the power. When finding the reciprocal of a fraction, the numerator and denominator swap. This means that 64 over 27 to the power of negative two-thirds is the same as 27 over 64 to the power of two-thirds. We also recall that when raising a fraction to a power, we can raise the numerator and denominator to the power separately. This means that we have 27 to the power of two-thirds over 64 to the power of two-thirds.

When raising any number to the power of 𝑚 over 𝑛, we firstly find the 𝑛th root of the number and then raise this answer to the power of 𝑚. To calculate 27 to the power of two-thirds, we find the cube root of 27 and then square our answer. The cube root of 27 is equal to three. As three squared is equal to nine, 27 to the power of two-thirds is equal to nine. We repeat this for the denominator. The cube root of 64 is equal to four. As four squared is equal to 16, 64 to the power of two-thirds is equal to 16. This means that 64 over 27 to the power of negative two-thirds is equal to nine over 16 or nine sixteenths.

In our next question, we will need to convert decimals into fractions first.

Evaluate 0.03125 to the power of negative 0.2.

In this question, our first step will be to convert the base number 0.03125 and our exponent negative 0.2 into fractions. 0.03125 is equal to the fraction 3125 over 100000. This is because the five is in the hundred thousandths column. It is not immediately obvious what the highest common factor of 3125 and 100000 is. We could begin by dividing the numerator and denominator by five, 25, 125, or any other factor of the numerator and denominator. 3125 does divide into 100000 32 times. This means that the fraction in its simplest form is one over 32. Our exponent in this question was negative 0.2, and this is the same as negative two-tenths. Dividing the numerator and denominator by two, this simplifies to negative one-fifth. We can, therefore, rewrite our calculation as one over 32 raised to the power of negative one-fifth.

We recall that raising the fraction 𝑎 over 𝑏 to the power of negative 𝑛 is the same as raising the fraction 𝑏 over 𝑎 to the power of 𝑛. This means we can rewrite our calculation as 32 over one to the power of one-fifth. Any integer divided by one is equal to the integer. Therefore, this could be rewritten as 32 to the power of one-fifth. Raising any number to the unit fraction one over 𝑛 is the same as finding the 𝑛th root of the number. We need to calculate the fifth root of 32. As two to the power of five is equal to 32, the fifth root of 32 is two. 0.03125 raised to the power of negative 0.2 is equal to two.

In our final question, we will combine all of the skills we have used in this video.

Evaluate four to the power of three over two multiplied by 32 to the power of negative 0.2 multiplied by two to the power of negative three all divided by 32 to the power of negative 0.4 multiplied by two to the power of negative two.

One way of approaching this problem would be to work out each of the five terms individually. We could work out four to the power of three over two, 32 to the power of negative 0.2, and so on. Whilst this would give us the correct answer, there is a quicker method if we recognize that four and 32 are powers of two. Four is equal to two squared and 32 is equal to two to the power of five. This means that we could rewrite our calculation as shown. Four to the power of three over two is the same as two squared to the power of three over two. 32 to the power of negative 0.2 is equal to two to the power of five to the power of negative 0.2. And 32 to the power of negative 0.4 is equal to two to the power of five to the power of negative 0.4.

We can now use our laws of exponents or indices. 𝑥 to the power of 𝑎 raised to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 multiplied by 𝑏. We can multiply two by three over two, five by negative 0.2, and five by negative 0.4. Two multiplied by three over two is equal to three. This means that the first term is equal to two cubed. Five multiplied by negative 0.2 is equal to negative one. Our second term is two to the power of negative one. The numerator is completed by multiplying by two to the power of negative three. Five multiplied by negative 0.4 is equal to negative two. This means that the numerator becomes two to the power of negative two multiplied by two to the power of negative two.

Another one of our laws of exponents states that 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏. We need to add three negative one and negative three on the numerator. This gives us negative one. So, we have two to the power of negative one. We repeat this on the denominator. Negative two plus negative two is equal to negative four. Our equation becomes two to the power of negative one over two to the power of negative four.

Our third rule of exponents states that 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏. We need to subtract negative four from negative one. This is the same as adding four to negative one, which gives us three. This leaves us with two to the power of three or two cubed. Two cubed is equal to eight. Therefore, this is the answer to our calculation.

Had we tried to solve the problem by working out each term individually, we could have used the rules that 𝑥 to the power of 𝑎 over 𝑏 is equal to the 𝑏th root of 𝑥 raised to the power of 𝑎 and 𝑥 to the power of negative 𝑎 is equal to one over 𝑥 to the power of 𝑎. Four to the power of three over two is equal to eight. 32 to the power of negative 0.2 or negative one-fifth is equal to one-half. This means that 32 to the power of negative 0.4 or negative two-fifths is equal to one-quarter. Two to the power of negative three is one-eighth and two to the power of negative two is one-quarter.

This would have left us with eight multiplied by a half multiplied by an eighth divided by a quarter multiplied by a quarter. The numerator simplifies to one-half and the denominator one sixteenth. As dividing a fraction by a fraction is the same as multiplying the first fraction by the reciprocal of the second, one-half divided by one sixteenth is also equal to eight. This confirms the answer we got using our first method.

We will now summarize the key points from this video. In this video, we recalled our three laws of exponents where we need to add, subtract, and multiply powers. When dealing with negative exponents, we saw that 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. We also learnt a rule for fractional exponents and used a combination of these to answer problems involving integers, fractions, and decimals.

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