List three different ways to write 5^11 as the product of two powers. Explain why all three of your expressions are equal to 5^11.
Can someone help me with this?
two powers of 5 whose exponents sum to 11
multiplying exponential quantities with the same base
... can be done by adding the exponents
Sure, I can help you.
Three different ways to write 5^11 as the product of two powers are:
1. (5^5) * (5^6)
2. (5^3) * (5^8)
3. (5^2) * (5^9)
To explain why all three of these expressions are equal to 5^11, we need to understand the properties of exponents.
The multiplication property of exponents states that when you multiply two numbers with the same base, you add their exponents. This property allows us to combine the powers of 5 in the expressions above.
1. In (5^5) * (5^6), we have the same base (5), so we can add the exponents: 5 + 6 = 11.
2. In (5^3) * (5^8), again, the base is the same (5), so we can add the exponents: 3 + 8 = 11.
3. Similarly, in (5^2) * (5^9), the base is 5, and the exponents can be added: 2 + 9 = 11.
Therefore, all three expressions (1, 2, and 3) represent the same value, which is 5^11.
Sure! I can help you with that.
To write 5^11 as the product of two powers, we need to find two exponents that can be multiplied together to give us 11. Here are three different ways to write 5^11 as the product of two powers:
1. 5^11 = 5^5 * 5^6: In this expression, we have 5 raised to the power of 5 and 5 raised to the power of 6. When we multiply these two powers together, we get 5^11.
2. 5^11 = 5^3 * 5^8: In this expression, we have 5 raised to the power of 3 and 5 raised to the power of 8. Again, when we multiply these two powers together, we get 5^11.
3. 5^11 = 5^2 * 5^9: In this expression, we have 5 raised to the power of 2 and 5 raised to the power of 9. Once again, when we multiply these two powers together, we get 5^11.
Now, let's explain why all three expressions are equal to 5^11. This is because of the exponentiation rule that states: When you multiply two powers with the same base, you need to add their exponents.
In all three expressions, we have the base 5 raised to different exponents, but when multiplied together, the exponents add up to 11. Therefore, all three expressions are equal to 5^11.