Describe the possible values of a such that √72 + √a simplifies to a single term.
I have seen several explanations on Jiskha & Slader, but I still don't understand all the steps all the way.
well, clearly, if a=72
√72 + √72 = 2√72
or, since √72 = 6√2, if a=2,
6√2 + √2 = 7√2
There are, of course, many more such values.
Well, let's see if I can shed some light on this while sprinkling a little humor along the way!
To simplify the expression √72 + √a, we want to find a value of a that will allow us to combine the square roots into a single term.
First, let's simplify the square root of 72. Now, 72 happens to be divisible by 36 and since 6 * 6 = 36, we can say that √72 = √(36 * 2). We can then pull out the 36 as it's a perfect square, giving us 6√2.
Now, to combine the square roots, we need to make sure that the number under the second square root, a, is divisible by 2. Why, you ask? Well, if it's not divisible by 2, the square root of a cannot be simplified any further with the square root of 72. So, a must be even.
That's it! So, the possible values of a such that √72 + √a simplifies to a single term are all even numbers.
Hope that explanation didn't leave you feeling square!
To simplify the expression √72 + √a into a single term, we need to simplify both square roots.
Let's start by simplifying the square root of 72. We can factorize 72 into its prime factors: 72 = 2^3 × 3^2.
Now, let's break down the square root of 72 using the property of square roots: √(a × b) = √a × √b.
√72 = √(2^3 × 3^2)
= √(2^3) × √(3^2)
= √8 × √9
Since the square root of 8 can simplify further, we can express it as the square root of 4 times the square root of 2.
√72 = √8 × √9
= √(4 × 2) × 3
= (2√2) × 3
= 6√2
Now, we have simplified the square root of 72 to 6√2.
Next, let's move on to simplifying the square root of a. Since we don't have any information about the value of a, we can't simplify it any further. Therefore, we leave it as √a.
Therefore, the expression √72 + √a simplifies to 6√2 + √a, which is a single term.
To find the possible values of a such that √72 + √a simplifies to a single term, we need to simplify the expression by combining the square roots.
Let's break it down step by step:
1. Start with the expression: √72 + √a
2. Since 72 is a perfect square, we can simplify √72 to a whole number. 72 can be factored as 8 * 9, and 9 is a perfect square, so we can rewrite it as 3²:
√72 = √(8 * 9) = √8 * √9 = 2√8 * 3
Therefore, our expression becomes: 2√8 * 3 + √a
3. Now, we can further simplify. √8 can be factored into √(4 * 2). Since 4 is a perfect square, we can rewrite it as 2²:
2√8 * 3 + √a = 2√(4 * 2) * 3 + √a = 2 * 2√2 * 3 + √a = 6√2 + √a
Now our expression is in simplified form.
To have the expression simplify to a single term, we need to eliminate the terms with square roots. This means that the value of a must be such that the term with the square root of a cancels out with the other term.
In this case, we have 6√2 + √a. To cancel the square roots, the coefficient of √2 (which is 6) must be equal to the coefficient of √a. Therefore, the value of a that satisfies this condition is a = 2² = 4.
By substituting a = 4 into our expression, we get:
6√2 + √4 = 6√2 + 2
Now we have a simplified expression with a single term.