Lesson Video: Simplifying Exponential Expressions with Integer Exponents Mathematics • Second Year of Preparatory School

Video Transcript

In this video, we will learn how to perform operations and simplifications on expressions that involve integer exponents. To help us understand how to do this, we will begin by recalling how to evaluate a power.

We know that for a power with base 𝑎, where 𝑎 is any real number apart from zero, and exponent 𝑛, where 𝑛 is a positive integer, then 𝑎 to the 𝑛th power is equal to 𝑎 multiplied by 𝑎 multiplied by 𝑎 and so on, where there are 𝑛 𝑎’s. For example, two to the fourth power is equal to two multiplied by two multiplied by two multiplied by two, which is equal to 16.

Next, we recall the rules for zero and negative exponents. If the base 𝑎 is any real number apart from zero, then 𝑎 to the power of zero is equal to one. When dealing with negative exponents, 𝑎 to the power of negative 𝑛 is equal to one over 𝑎 to the 𝑛th power, where 𝑎 is any real number apart from zero. In this video, we will only consider integer values of 𝑛. However, it is important to note that this holds for all values of 𝑛. Likewise, 𝑎 to the 𝑛th power is equal to one over 𝑎 to the power of negative 𝑛. In our first example, we will use these definitions to simplify a numerical expression with integer exponents.

Calculate 10 to the fifth power multiplied by 25 squared multiplied by four divided by 20 squared multiplied by five to the fifth power.

To calculate this expression, we can write the powers in expanded form and then cancel common factors in the numerator and denominator. The first term in the numerator, 10 to the fifth power, can be rewritten as 10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10. 25 squared can be written as 25 multiplied by 25. This means that the entire numerator can be rewritten as shown.

We can do the same with the denominator and rewrite 20 squared and five to the fifth power. Whilst we could cancel at this stage, we can simplify our expression further by writing the numerator and denominator as products of prime factors. We know that 10 is equal to two multiplied by five. 25 can be rewritten as five multiplied by five. We can rewrite four as two multiplied by two. And 20 written as a product of prime factors is two multiplied by two multiplied by five.

We can use the first three of these to rewrite the numerator as shown. And repeating this process with the denominator, we have the following expression. Next, we can cancel common factors. Two multiplied by two multiplied by two multiplied by two can be canceled from the numerator and denominator. We can also cancel seven fives from the numerator and denominator. Our expression therefore simplifies to two multiplied by five multiplied by five multiplied by two multiplied by two all divided by one. This is equal to two cubed multiplied by five squared. And since two cubed is equal to eight and five squared is 25, we have eight multiplied by 25, which is equal to 200. 10 to the fifth power multiplied by 25 squared multiplied by four divided by 20 squared multiplied by five to the fifth power is 200.

Sometimes when simplifying expressions with integer exponents, we need to use the laws of exponents to do this. Before looking at our next example, we will recall these laws of exponents. In this video, we will only consider the following laws when we have nonzero bases and integer exponents. Firstly, we have the product rule, which states that 𝑎 to the power of 𝑚 multiplied by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 plus 𝑛. Next, we have the quotient rule. This states that 𝑎 to the power of 𝑚 divided by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 minus 𝑛. The power of a product rule states that 𝑎𝑏 to the power of 𝑛 is equal to 𝑎 to the power of 𝑛 multiplied by 𝑏 to the power of 𝑛. In a similar way, the power of a quotient rule states that 𝑎 over 𝑏 to the power of 𝑛 is equal to 𝑎 to the power of 𝑛 divided by 𝑏 to the power of 𝑛. Finally, we have the power rule, which states that 𝑎 to the power of 𝑚 raised to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 multiplied by 𝑛.

We will now consider a couple of examples where we need to use these laws of exponents.

Simplify four to the power of 𝑥 multiplied by four to the power of four 𝑥 divided by four to the power of three 𝑥 minus four multiplied by 16 to the power of 𝑥.

In order to simplify this expression, we will need to use the laws of exponents. We recall that in order to use these, each term must have the same base. In this question, three of the four terms have a base of four. Our first step is therefore to rewrite 16 to the power of 𝑥 with a base of four. We know that four squared is equal to 16. This means that we can rewrite 16 to the power of 𝑥 as four squared to the power of 𝑥. The initial expression can therefore be rewritten as four to the power of 𝑥 multiplied by four to the power of four 𝑥 divided by four to the power of three 𝑥 minus four multiplied by four squared to the power of 𝑥.

We will now consider three of the laws of exponents. Firstly, the product rule states that 𝑎 to the power of 𝑚 multiplied by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 plus 𝑛. This means that the numerator can be rewritten as four to the power of 𝑥 plus four 𝑥. Before applying the same rule to the denominator, we need to consider the power rule of exponents. This states that 𝑎 to the power of 𝑚 all raised to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 multiplied by 𝑛. As such, four squared raised to the power of 𝑥 can be rewritten as four to the power of two 𝑥.

We can then use the product rule to rewrite the denominator as four to the power of three 𝑥 minus four plus two 𝑥. 𝑥 plus four 𝑥 is equal to five 𝑥, and three 𝑥 minus four plus two 𝑥 is equal to five 𝑥 minus four. So our expression simplifies to four to the power of five 𝑥 divided by four to the power of five 𝑥 minus four.

We can now use the quotient rule, which states that 𝑎 to the power of 𝑚 divided by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 minus 𝑛. Subtracting five 𝑥 minus four from five 𝑥, we have four to the power of five 𝑥 minus five 𝑥 minus four. Distributing the parentheses, the exponent becomes five 𝑥 minus five 𝑥 plus four. And our expression simplifies to four to the fourth power. Evaluating this, we get four multiplied by four multiplied by four multiplied by four, which is equal to 256. Four to the power of 𝑥 multiplied by four to the power of four 𝑥 divided by four to the power of three 𝑥 minus four multiplied by 16 to the power of 𝑥 is equal to 256.

In our next example, we will use the laws of exponents to simplify an algebraic expression with two different bases.

Simplify 𝑥 to the power of 𝑛 plus nine multiplied by 𝑦 to the power of 𝑛 plus four divided by 𝑥 to the power of 𝑛 plus three multiplied by 𝑦 to the power of 𝑛.

In order to simplify this expression, we can use the laws of exponents. However, we notice that we have two different bases. Since the laws only apply to exponents with the same bases, we can rewrite the expression as shown. We have 𝑥 to the power of 𝑛 plus nine divided by 𝑥 to the power of 𝑛 plus three multiplied by 𝑦 to the power of 𝑛 plus four divided by 𝑦 to the power of 𝑛. Recalling that the quotient rule for exponents states that 𝑎 to the power of 𝑚 divided by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 minus 𝑛, we can rewrite the first part of our expression as 𝑥 to the power of 𝑛 plus nine minus 𝑛 plus three.

In the same way, the second part of our expression simplifies to 𝑦 to the power of 𝑛 plus four minus 𝑛. In both parts, the 𝑛’s cancel. And since nine minus three is equal to six, we have 𝑥 to the power of six multiplied by 𝑦 to the power of four. The expression 𝑥 to the power of 𝑛 plus nine multiplied by 𝑦 to the power of 𝑛 plus four divided by 𝑥 to the power of 𝑛 plus three multiplied by 𝑦 to the power of 𝑛 written in its simplest form is 𝑥 to the sixth power 𝑦 to the fourth power.

We will now look at one final example where we need to simplify an algebraic expression using the laws of exponents and by taking out common factors that are powers.

Find the value of 𝑎 for which two to the power of 𝑥 plus six minus two to the power of 𝑥 plus two is equal to 𝑎 multiplied by two to the power of 𝑥.

In order to find the value of 𝑎 in this equation, we need the left-hand side of the equation to be in the form of the right-hand side. In other words, we need to rewrite two to the power of 𝑥 plus six minus two to the power of 𝑥 plus two so it is in the form 𝑎 multiplied by two to the power of 𝑥.

We recall that the product rule of exponents states that 𝑎 to the power of 𝑚 multiplied by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 plus 𝑛. This means that we can rewrite the first term as two to the power of 𝑥 multiplied by two to the power of six. Likewise, two to the power of 𝑥 plus two can be rewritten as two to the power of 𝑥 multiplied by two squared. Both of our terms now have a common factor of two to the power of 𝑥.

Factoring this out, we have two to the power of 𝑥 multiplied by two to the sixth power minus two squared. We know that two squared is equal to four and two to the sixth power is 64. As such, our expression simplifies to two to the power of 𝑥 multiplied by 60 or 60 multiplied by two to the power of 𝑥. Since this is equal to 𝑎 multiplied by two to the power of 𝑥, we can compare the coefficients, giving us a value of 𝑎 equal to 60.

We will now finish this video by summarizing the key points. We saw in this video that we can simplify exponential expressions by calculating the value of powers and canceling down common factors. We can also simplify exponential expressions using the laws of exponents. We have the product rule for exponents, which states that 𝑎 to the power of 𝑚 multiplied by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 plus 𝑛. The quotient rule states that 𝑎 to the power of 𝑚 divided by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 minus 𝑛. Thirdly, the power rule of exponents states that 𝑎 to the power of 𝑚 raised to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 multiplied by 𝑛. We also have the power of a product rule, which states that 𝑎𝑏 to the power of 𝑛 is equal to 𝑎 to the power of 𝑛 multiplied by 𝑏 to the power of 𝑛, and the power of a quotient rule, which states that 𝑎 over 𝑏 to the power of 𝑛 is equal to 𝑎 to the power of 𝑛 divided by 𝑏 to the power of 𝑛.

In this video, we only considered these for nonzero bases and integer exponents. However, it’s important to note that they also hold for noninteger exponents. We also saw in our last example that we can use the laws of exponents to solve equations.