Question #94618

2 Answers
Dec 25, 2017

sqrt3900=10sqrt39

Approximate form: 62.44997998 larr using calculator

Explanation:

Prime factorize 3900.

sqrt3900=sqrt(2xx2xx3xx5xx5xx13)

Group the same prime factors into pairs.

sqrt3900=sqrt((2xx2)xx(5xx5)xx3xx13)

Rewrite each pair in exponent form.

sqrt3900=sqrt(2^2xx5^2xx3xx13)

Apply the rule: sqrt(x^2)=x

sqrt3900=(2xx5)sqrt(3xx13)

Simplify.

sqrt3900=10sqrt39

Dec 25, 2017

Simplify the square root (by "splitting" then solving), and approximate if needed:

sqrt(3900) = 10 sqrt(39) ~~ 10 * 6.245 = 62.45

Explanation:

A rule about exponents is that raising a product to a power, is the same as taking the product of each multiplier having been raised to the same power:

(ab)^c = a^c b^c

And the square root of a number can be rewritten as the same number raised to an exponent of 1/2:

sqrt(n) = n^(1/2)

Why is that? Well, when we say square root, we mean a number where:

sqrt(n) * sqrt(n) = n

n = n^1 and a number times itself is the number squared (a * a = a^2), so:

(sqrt(n))^2 = n^1

So if we rewrite sqrt(n) as n^p (where we don't yet know the value of p), we get:

(n^p)^2 = n^1

And (a^b)^c = a^(bc):

n^(2p) = n^1

Shouldn't it be that 2p = 1 rarr p = 1/2?

If (ab)^c = a^c b^c and sqrt(n) = n^(1/2), then...

sqrt(ab) = (ab)^(1/2) = a^(1/2) b^(1/2) = sqrt(a) * sqrt(b)

This rule is important here! It tells us that we can somehow "split" the square root of a product!

Now back to our problem:

sqrt(3900) = "???"

Well, 3900 = 39 * 100, so:

sqrt(3900) = sqrt(39) * sqrt(100) = 10 sqrt(39)

Unfortunately, we can't simplify this any further, but we can approximate sqrt(39):

sqrt(39) ~~ 6.245

So:

sqrt(3900) = 10 sqrt(39) ~~ 10 * 6.245 = 62.45

Pay attention to the squiggly lines; it indicates an approximation! So if they ask for an exact value, just give 10 sqrt(39), but you may use the approximation in solving word problems.