Gr9 Maths: Exponents | Term 1 | Lesson 5 | Calculations Using Exponents

Added: 2026-05-13

[00:00:06]Hello grade nines. Welcome to tumina teaching. I am novela and this is your last maths lesson in our series about exponents.

[00:00:18]In this lesson, we are simplifying expressions using the laws of exponents that you already know. Because these expressions mix powers, roots, multiplication or division, the order of operations helps us to decide what to simplify first.

[00:00:40]Remember, mathematics is built step by step. This includes knowledge, the ideas and concepts that you are learning, skills, how you solve the problems, and fluency, how easily and confidently you can do the maths with practice.

[00:01:01]Make sure you understand exponents well because the mathematics you do in the next grade builds on this knowledge.

[00:01:10]Okay, let's do this. grade nines.

[00:01:18]Let's start with this expression.

[00:01:22]We have a product and the first part is a power of a product.

[00:01:28]Remember when a bracket is raised to a power, we multiply the exponent inside the bracket with the power outside the bracket. This is the power of a product law.

[00:01:41]So a raised to the power of 2 and b to the power of 1 raised to the power of 2 becomes a to the power of 2 * 2 and b to the power of 1 * 2.

[00:02:04]Next we multiply the a to the power of 4 b to the power of 2 by a to the power of 3 b to the power of zero.

[00:02:20]Remember anything to the power of 0 is 1. So b to the power of zero won't change the answer.

[00:02:32]Now we use the product law.

[00:02:36]When we multiply the same bases, we add the exponents. So we add the exponents for each base. A to the power of 4 + 3 b to the power of 2 + 0 and get a to the power of 7 b to the power of 2.

[00:03:01]Grade nines, if you are working with a friend, try explaining the idea to each other. Only when you understand something well, can you explain it to others in a way that they can understand.

[00:03:15]This is a good way to check your own understanding of a concept. If this step still feels tricky, pause here and look at how the exponent law was used. Replay this part if you need to.

[00:03:37]Let's simplify this expression step by step.

[00:03:41]We begin with the numerical coefficients.

[00:03:44]12 / 4 gives us 3. Before we combine the exponents, let's rewrite the expression using positive exponents only. Remember the negative exponent law. A negative exponent tells us that we can rewrite the base on the other part of the fraction with a positive exponent.

[00:04:11]So rewriting with positive exponents, we now have x ^ of 4 * x ^ of 2 over y ^ of 3 * y ^ of 1. We can see x ^ of -2 is now written on the numerator with a positive exponent and y to the power of -1 is written on the denominator with a positive exponent.

[00:04:46]Now we use the product law of exponents.

[00:04:50]When we multiply expressions with the same base, we keep the base the same and add the exponents.

[00:04:59]For the x terms, 4 + 2 gives us six. And for the y terms, 3 + 1 gives 4.

[00:05:11]So our simplified expression is 3 x ^ of 6 over y ^ of four all with positive exponents. Take note writing the numerical coefficient in front of the fraction is the same as writing it in the numerator.

[00:05:37]Let's look at this example.

[00:05:40]3 * 1 / a. We can write 3 as a fraction.

[00:05:48]3 over 1 then 3 over 1 * 1 / a which equals 3 / a. This shows that writing the coefficient in front or in the numerator gives us the same results.

[00:06:14]In this example, we have a square root and a cube root. So, we'll simplify each part before multiplying.

[00:06:28]We begin with the square root. The square root of 16 is 4. And for p to the power of 8, we divide the exponent by 2, which gives us p to the power of 4. So the square root simplifies to 4 p to the power of 4.

[00:06:51]Now let's simplify the cube root.

[00:06:59]The cube root of a negative exponent is allowed because cube roots work with all real numbers. And we know that p to the power of -2 cubed will be p to the power of -6.

[00:07:19]For p to the power of -6, we divide the exponent by 3, which gives us p to the power of -2.

[00:07:32]And for q to the power of 9, dividing the exponent by 3 gives us q to the so the cube root becomes p to the power of -2 q to the power of 3. Next, we multiply the terms. For like bases, we add the exponents.

[00:08:00]So for the p terms 4 + -2, which is 4 - 2 and q to the^ of 3 stays as it is. So our simplified answer is 4 p to ^ of 2 q to the power of 3.

[00:08:28]Let's start by writing each fraction using positive exponents only. This makes it much easier to apply the exponent laws. In this fraction for m m to the power of -1 in the denominator we move it to the top so the exponent becomes positive m to the power of 1.

[00:08:55]For n to the power of -2 in the numerator we move it down so it becomes n to the power of two.

[00:09:06]Now we do the same for this fraction.

[00:09:09]The negative exponent is m to the power of -2 in the numerator.

[00:09:17]So by the negative exponent law we move it to the denominator and write it as m to the power of two.

[00:09:28]Now that all the exponents are positive, we can apply the product law. In the first fraction, we multiply m to the power of 3 and m to the power of 1. So we add their exponents 3 + 1. In the denominator, we multiply n to the power of 4 and n to the power of 2. So we add those exponents as well. 4 + 2. In the second fraction, 10 / 2 gives us 5. m to the power of 3 and m to the power of 2 are both in the denominator. So we add their exponents. 3 + 2.

[00:10:22]n ^ of two stays n ^ 2 in the numerator.

[00:10:28]After adding the exponents, we get 5 m ^ of 4 over n ^ of 6 ultiplied by 5 n ^ of 2. Step one, multiply the numerators and the denominators.

[00:10:49]Step two, multiply the coefficients.

[00:10:57]5 * 5 gives us 25. So we get 25 m to ^ of 4 n ^ of 2 over n ^ of 6 m. Step three, use the quotient law and subtract the exponents.

[00:11:22]For m 4 - 5 gives m to the power of -1.

[00:11:29]For n 2 - 6 gives us m to the power of step four rewrite with negative exponents.

[00:11:41]Step five, move the negative exponents to the denominator to make them positive.

[00:11:49]So our final answer with positive exponents only is 25 over m n^ of 4.

[00:12:01]Red nines, we're using several exponent laws at once here. Take your time, pause, and check again if you need to.

[00:12:10]Also think of how you would explain this to someone else.

[00:12:22]Well, now it's your turn.

[00:12:26]Pause the video, take out your notebook, and try these five questions. Grade nines, work carefully. Use the exponent laws you've learned and rewrite your final answers with positive exponents only. If you feel stuck at any question, pause and look back at the other examples.

[00:12:48]Ask yourself, are the bases the same? Am I multiplying or dividing or raising a power to a power? Should any negative exponent be rewritten on the other side of the fraction? Remember, you can replay the explanation, slow it down, or try the question again. That's how you learn.

[00:13:18]Let's check the answers together. Pause and compare your work carefully.

[00:13:37]Where are you?

[00:13:46]Where are you?

[00:14:13]Where are you?

[00:14:38]Look closely at number four. Remember that n the power of 3 - -2 is the same as n to the power of 3 + 2 which gives us n to the power of 5.

[00:15:35]Well, grade nines, this brings us to the end of this lesson.

[00:15:40]and also to the end of our series about exponents.

[00:15:49]Remember, you can always pause, replay, and practice these steps again. Don't forget to test yourself after the lesson. Click on the link and complete the selfmarking assessment in the description below. Wherever you're learning from, don't forget to subscribe to our channel so that you don't miss out on our next lesson.

[00:16:14]See you again, grade nines.