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
1. 5^5 x 5^6: This expression breaks down 5^11 into 5 multiplied by itself 5 times (5^5) and then multiplied by itself 6 more times (5^6).
2. 5^10 x 5^1: This expression breaks down 5^11 into 5 multiplied by itself 10 times (5^10) and then multiplied by itself one more time (5^1).
3. (5^3)^3 x 5^2: This expression breaks down 5^11 into 5 multiplied by itself 3 times, and then that result multiplied by itself 3 times (5^3)^3, and then multiplied by itself 2 more times (5^2).
All three of these expressions are equal to 5^11 because they all represent the same basic idea: multiplying 5 by itself 11 times. Each of the expressions just breaks down that idea in a slightly different way using the rules of exponents.
Sure! Here are all three expressions together:
1. 5^5 x 5^6
2. 5^10 x 5^1
3. (5^3)^3 x 5^2
All of these expressions are equal to 5^11 because they all describe multiplying 5 by itself 11 times, but they break it down in different ways using the rules of exponents.
can you put the anwsers all together
To write 5^11 as the product of two powers, we need to express it in a form where the base and the exponent are multiplied. Here are three different ways to write 5^11 as the product of two powers:
1. (5^5) * (5^6)
2. (5^2) * (5^9)
3. (5^3) * (5^8)
Now let's explain why all three expressions are equal to 5^11:
1. (5^5) * (5^6) = 5^(5 + 6) = 5^11
Here, we use the rule of exponentiation that states: a^m * a^n = a^(m + n). By adding the exponents of 5, we get 11, so the expression is equal to 5^11.
2. (5^2) * (5^9) = 5^(2 + 9) = 5^11
Similar to the first expression, we add the exponents of 5, which results in 11. Therefore, the expression is equal to 5^11.
3. (5^3) * (5^8) = 5^(3 + 8) = 5^11
Once again, by combining the exponents of 5, we obtain 11. Thus, the expression is equal to 5^11.
In all three cases, the base 5 remains the same, and the sum of the exponents is 11, which means that the three expressions are indeed equal to 5^11.