Can you provide an explanation for why adding exponents is fundamental when rewriting 5 to the power of 11 as the product of two powers?
Certainly! When rewriting a number as a product of two powers, we can use the rule of exponents that states:
a^m * a^n = a^(m+n)
Now, let's apply this rule to the number 5^11. We can rewrite it as:
5^11 = 5^5 * 5^6
Here, we used the rule of exponents to split 11 into 5 and 6 (11 = 5 + 6).
To understand why we can add the exponents in this way, let's break it down:
When we raise a number to an exponent, we are essentially multiplying that number by itself repeatedly. In the case of 5^11, we multiply 5 by itself eleven times.
Now, when we multiply two powers with the same base, like 5^5 and 5^6, we are essentially multiplying the base (5) by itself a total of 5 + 6 = 11 times.
So, by adding the exponents, we are actually combining the powers by multiplying the base repeatedly. In the example above, we have 5^5 * 5^6, which gives us 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5.
Therefore, adding exponents is fundamental when rewriting 5^11 as the product of two powers, as it allows us to express the repeated multiplication of the base in a condensed form.
Certainly! When rewriting 5 to the power of 11 as the product of two powers, we can break it down into smaller exponent expressions. Let's see step-by-step how this is done:
Step 1: The expression we are starting with is 5 to the power of 11. (5^11)
Step 2: We want to write this expression as the product of two powers. This means that we need to split the exponent into two parts.
Step 3: We can rewrite the exponent 11 as a sum of two smaller exponents. Since 11 is an odd number, we can split it into 10 and 1, which are 10 and 1.
Step 4: Now, we can express 5 to the power of 11 as follows:
5^11 = (5^10) * (5^1).
Step 5: To simplify this expression, we use the property of exponents that states when multiplying two powers with the same base, we add their exponents.
Therefore, (5^10) * (5^1) = 5^(10+1).
Step 6: Adding the exponents gives us 5^11.
Therefore, (5^10) * (5^1) = 5^11.
This process allows us to rewrite 5 to the power of 11 as the product of two powers by utilizing the fundamental property of exponents which states that adding exponents is equivalent to multiplying powers with the same base.