We know how to calculate the expression 5 x 5. This expression can be written in a shorter way using something called exponents. $$5\cdot 5=5^{2}$$ An expression that represents repeated multiplication of the same factor is called a power. The number 5 is called the base, and the number 2 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.  Example Write these multiplications like exponents $$5\cdot 5\cdot 5=5^{3}$$ $$4\cdot 4\cdot 4\cdot 4\cdot 4=4^{5}$$ $$3\cdot 3\cdot 3\cdot 3=3^{4}$$ MultiplicationIf two powers have the same base then we can multiply the powers. When we multiply two powers we add their exponents. The rule: $$x^{a}\cdot x^{b}=x^{a+b}$$ Example $$4^{2}\cdot 4^{5}=\left ( 4\cdot 4 \right )\cdot \left ( 4\cdot 4\cdot 4\cdot 4\cdot 4 \right )=4^{7}=4^{2+5}$$ DivisionIf two powers have the same base then we can divide the powers. When we divide powers we subtract their exponents. The rule: $$\frac{x^{a}}{ x^{b}}=x^{a-b}$$ Example $$\frac{4^{2}}{ 4^{5}}=\frac{{\color{red} {\not}{4}}\cdot {\color{red} {\not}{4}}}{{\color{red} {\not}{4}}\cdot {\color{red} {\not}{4}}\cdot 4\cdot 4\cdot 4}=\frac{1}{4^{3}}=4^{-3}=4^{2-5}$$ A negative exponent is the same as the reciprocal of the positive exponent. $$x^{-a}=\frac{1}{x^{a}}$$ Example $$2^{-3}=\frac{1}{2^{3}}$$ When you raise a product to a power you raise each factor with a power $$(x\cdot y)^{a}=x^{a}\cdot y^{a}$$ Example $$(2x)^{4}=2^{4}\cdot x^{4}=16x^{4}$$ The rule for the power of a power and the power of a product can be combined into the following rule: $$(x^{a}\cdot y^{b})^{z}=x^{a\cdot z}\cdot y^{b\cdot z}$$ Example $$(x^{3}\cdot y^{4})^{2}=x^{3\cdot 2}\cdot y^{4\cdot 2}=x^{6}\cdot y^{8}$$ Video lessonsRewrite the expressions $$2\cdot 2\cdot 2$$ $$x\cdot x\cdot x\cdot x\cdot x$$ $$3^{4}$$ $$x^{3}$$ Simplify the expression $$\left ( x^{2}\cdot y^{3}\cdot z^{5} \right )^{3}$$
In this explainer, we will learn how to identify the base and exponent in power formulas, write them in exponential, expanded, and word forms, and evaluate simple powers. A repeated addition can be written as a multiplication. So, if we get 10 messages one day, 10 the next day, and 10 the day after, we have got, in total, 10+10+10=3×10 messages. The repeated addition of the number 10 is written as a multiplication of 10 by the number of times 10 appears in the repeated addition. (We could also say it is one more than the number of times 10 is added to itself.) Similarly, there is a way to write repeated multiplication. For instance, let us suppose that someone receives a picture on social media. This person shares it in the first minute to four of his or her friends. In the following minute, each of the four friends shares it to another four people. And, in the third minute, each of the people who received the picture in the previous minute shares it to another four people. The number of people who got this picture in the third minute is 1⋅4⋅4⋅4. This repeated multiplication of a given factor (here 4) can be written as a power of this factor; namely, 1⋅4⋅4⋅4=4=64. The number 4 is called a power of 4. The factor 4 that is repeated is called the base, and the exponent 3 is the number of times the factor appears in the repeated multiplication. (We could also say it is one more than the number of times 4 is multiplied by itself.) Let us summarize what we have just learned about powers. Powers are numbers resulting from a repeated multiplication of a factor. Their general form is 𝑏, where 𝑏 is called the base and 𝑚 the exponent. The base 𝑏 is the factor repeatedly multiplied by itself, and the exponent 𝑚 is the number of times 𝑏 appears in the repeated multiplication. When a number is written as a power, we say it is written in exponential form. When a power is written as a repeated multiplication, we say it is written in expanded form. You may have already learned about the square and cube of a number. Squaring is the same as raising to the second power; it is multiplying a number by itself: 𝑎=𝑎×𝑎. A number raised to the third power, or the power of three, is cubed: 𝑎=𝑎×𝑎×𝑎. Note that any number can be written as the first power of itself, 3=3. The identity property multiplication tells us that multiplying any number by 1 does not change this number. It follows that all the powers of 1 are simply 1: 1=1, 1=1, 1=1, 1=1, and so on. Also, raising any nonzero number to the zeroth power gives 1. Let us go through several examples to check and deepen our understanding. Write 7×7×7×7 in exponential form. AnswerThe expression 7×7×7×7 is a repeated multiplication of the factor 7. It can be written in exponential form, that is, in the form 𝑏, where 𝑏 is the factor that is repeatedly multiplied by itself, here 7, and 𝑚 is the number of times this factor appears in the repeated multiplication, here 4. Hence, 7×7×7×7=7. Write 3 in expanded form. AnswerA power is written in the form 𝑏. It is a shorthand for a repeated multiplication of a factor 𝑏, with 𝑚 being the number of times this factor appears in the repeated multiplication. Here, 𝑏=3 and 𝑚=4. Hence, 3=3×3×3×3. Write seven to the fourth power using digits. AnswerWe are asked to write seven to the fourth power. Recall that a power is written in the form 𝑏. It is a shorthand for a repeated multiplication of a factor 𝑏, with 𝑚 being the number of times this factor appears in the repeated multiplication. There are different ways to call 𝑏: 𝑏 raised to the 𝑚th power or to the power of 𝑚, or 𝑏 to the 𝑚th power. Here, we have seven to the fourth power, so we find that 𝑏=7 and 𝑚=4. Hence, it is written as 7. Express 3 as a product of the same factor, and then find its value. AnswerThe number 3 is written in exponential form and we want first to write it in expanded form. The exponential form 𝑏 is a shorthand for a repeated multiplication of a factor 𝑏, with 𝑚 being the number of times this factor appears in the repeated multiplication. Hence, 3 in expanded form is 3⋅3⋅3⋅3. Then, we need to evaluate this product. Using associativity, we can write 3⋅3⋅3⋅3=(3⋅3)⋅(3⋅3)=9⋅9=81. Let us look now at an example of multiplications of powers of different bases. Which of the following is equivalent to 7⋅10? AnswerThe expression 7⋅10 involves the multiplication of two powers, 7 and 10. Let us expand each of them. Recall that a power, written in the form 𝑏, is a shorthand for a repeated multiplication of a factor 𝑏, with 𝑚 being the number of times this factor appears in the repeated multiplication. Hence, we have 7=7⋅7⋅7⋅7 and 10=10⋅10⋅10⋅10⋅10⋅10. Now, we simply multiply them together to find an expression equivalent to 7⋅10: 7⋅7⋅7⋅7⋅10⋅10⋅10⋅10⋅10⋅10. Let us look at another example to understand what happens when two powers of the same base are multiplied. In this example, we have a power of a variable, 𝑥, instead of a number, and we will see that it does not change anything to the way we handle powers. Simplify 𝑥×𝑥. AnswerWe want to express 𝑥×𝑥 as a single power. To visualize better how this expression can be simplified, let us first rewrite it by expanding both powers. We find that our expression is (𝑥⋅𝑥⋅𝑥⋅𝑥⋅𝑥)⋅(𝑥⋅𝑥). The parentheses can be removed here using the associative property of multiplication. This repeated multiplication of the factor 𝑥 involves in total 5+2=7𝑥s. Hence, it can be written as 𝑥. We have seen in the previous two examples that when we multiply two powers of the same base, we get This is known as the product rule. The product of two powers that have the same base is a power of this same base with an exponent equal to the sum of the exponents: 𝑏⋅𝑏=𝑏.() In the following two examples, we are going to see how the commutative property of multiplication is used to rewrite expressions involving repeated multiplications of different factors. Which of the following expressions is equivalent to 7⋅5⋅7⋅3⋅5⋅5? AnswerThe expression 7⋅5⋅7⋅3⋅5⋅5 is a multiplication involving different factors. Therefore, it cannot be expressed as a single power. However, we notice that the factors 7 and 5 are repeated. We are going to use the commutative property of multiplication to first rewrite our expression so that all identical factors are grouped together. For instance, we may rewrite 7⋅5⋅7⋅3⋅5⋅5 as 7⋅7⋅3⋅5⋅5⋅5. It is now easy to see that 7 is used twice, 3 once, and 5 thrice. Recall that a power, written in the form 𝑏, is a shorthand for a repeated multiplication of a factor 𝑏, with 𝑚 being the number of times this factor appears in the repeated multiplication. Hence, this expression is equivalent to 7⋅3⋅5. Using commutativity again, we can change the order of the factors and write it as 3⋅5⋅7. Which of the following expressions is equivalent to 2⋅5⋅2⋅2⋅5⋅2⋅5⋅5? AnswerThe expression 2⋅5⋅2⋅2⋅5⋅2⋅5⋅5 is a multiplication involving different factors. Therefore, it cannot be expressed as a single power. However, we notice that the factors 2 and 5 are repeated. Using the commutative property of multiplication, we can rewrite our expression by grouping the twos together and the fives together. We get 2⋅2⋅2⋅2⋅5⋅5⋅5⋅5. Recall that a power, written in the form 𝑏, is a shorthand for a repeated multiplication of a factor 𝑏, with 𝑚 being the number of times this factor appears in the repeated multiplication. Hence, our expression is equivalent to 2⋅5. Still using the commutative property of multiplication, we can also rewrite our first expression as 2⋅5⋅2⋅5⋅2⋅5⋅2⋅5, which, using the associative property this time, is equivalent to (2⋅5)⋅(2⋅5)⋅(2⋅5)⋅(2⋅5). We see that the number inside each pair of brackets is 2⋅5, that is, 10. Hence, our expression is also equivalent to 10⋅10⋅10⋅10, which can be written as the fourth power of 10: 10. When we evaluate 10, we find it is 10 000. In conclusion, we have found that 2⋅5⋅2⋅2⋅5⋅2⋅5⋅5=2⋅5=10=10000. Therefore, the correct answer is option A: 2⋅5. In this last example, note that once you have understood the method, you do not need to group all the identical factors. You can simply count the number of times each factor appears and then write an equivalent power to the repeated multiplication of each factor. We have found in the previous two examples that when raising a product of two factors to the 𝑚th power, we have This can be summarized in this way. A given power of a product of factors is the product of each factor raised to that given power: (𝑎𝑏)=𝑎⋅𝑏. What is 411×411×411×411×411×411×411? AnswerAn exponent represents the number of times a rational number is multiplied by itself. In this expression, 411 is multiplied by itself 7 times. So, 411×411×411×411×411×411×411=411.
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What is the term for a number that indicates a value is multiplied by itself one or more times?
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