write equivalent expressions for x^7 ⋅ x^-2 and x^7 ÷ x^2.
What do you notice? Explain how your results relate to the properties of integer exponents.
THANK YOU!
The equivalent expressions for x^7 ⋅ x^-2 can be simplified using the properties of integer exponents. The property states that when multiplying variables with the same base, you add their exponents. Using this property, we can rewrite the expression as x^(7 + (-2)):
x^7 ⋅ x^-2 = x^(7 + (-2))
Simplifying the exponent, 7 + (-2) equals 5:
x^7 ⋅ x^-2 = x^5
The equivalent expression for x^7 ÷ x^2 can be obtained by using the property that states when you divide variables with the same base, you subtract their exponents. Applying this property, we can rewrite the expression as x^(7 - 2):
x^7 ÷ x^2 = x^(7 - 2)
Simplifying the exponent, 7 - 2 equals 5:
x^7 ÷ x^2 = x^5
Both expressions, x^7 ⋅ x^-2 and x^7 ÷ x^2, simplify to x^5. This shows that regardless of whether we multiply or divide variables with the same base, the result has an exponent equal to the difference between the exponents of the base. In this case, the result is x^5. This illustrates the power of the properties of integer exponents, which allow us to simplify expressions and apply consistent rules when dealing with multiplication and division of variables with the same base.
To find the equivalent expressions for the given expressions, we need to simplify them using the properties of integer exponents.
Expression 1: x^7 ⋅ x^-2
According to the property of exponent addition, when we multiply two numbers with the same base, we add the exponents. So, in this case, we can add 7 and -2 to get 7 + (-2) = 5.
Therefore, x^7 ⋅ x^-2 = x^5.
Expression 2: x^7 ÷ x^2
According to the property of division of exponents, when we divide two numbers with the same base, we subtract the exponents. So, in this case, we can subtract the exponent of the denominator from the exponent of the numerator. Thus, 7 - 2 = 5.
Therefore, x^7 ÷ x^2 = x^5.
What we notice is that both expressions simplify to x^5. This demonstrates the power of the properties of integer exponents.
The result relates to the property of exponent addition and division. When we multiply two numbers with the same base and different exponents (x^a ⋅ x^b), we add the exponents (a + b). Similarly, when we divide two numbers with the same base and different exponents (x^a ÷ x^b), we subtract the exponents (a - b). In this case, both expressions simplified to x^5, proving that these properties hold true.
These properties of integer exponents provide a systematic and efficient way to simplify expressions involving exponents, making calculations easier and allowing us to combine like terms conveniently.
I am doing the second part first :)
x^a * x^b = x^(a+b)
and
x^-a = 1/x^a
therefore
x^7 * x^-2 = x^7/x^2 = x^5
or
x^7 * x^-2 = x^(7-2) = x^5
remarkable :)
x^7 / x^2 = x^7 * x^-2
which we already did :)