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Let’s check out your problem:
both positive, write the followi
\newline
y
3
45
x
7
y
5
y^{3} \sqrt{45 x^{7} y^{5}}
y
3
45
x
7
y
5
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Math Problems
Calculus
Gaussian integral
Full solution
Q.
both positive, write the followi
\newline
y
3
45
x
7
y
5
y^{3} \sqrt{45 x^{7} y^{5}}
y
3
45
x
7
y
5
Break down square root:
First, let's break down the square root of
45
45
45
into prime factors:
45
=
9
×
5
=
9
×
5
=
3
5
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}
45
=
9
×
5
=
9
×
5
=
3
5
.
Apply exponent properties:
Now, let's apply the properties of exponents to the
x
x
x
and
y
y
y
terms inside the square root:
x
7
y
5
=
x
7
2
y
5
2
\sqrt{x^{7}y^{5}} = x^{\frac{7}{2}}y^{\frac{5}{2}}
x
7
y
5
=
x
2
7
y
2
5
.
Combine coefficients and terms:
Combine the coefficients and the like terms:
y
3
×
3
5
×
x
7
2
y
5
2
y^{3} \times 3\sqrt{5} \times x^{\frac{7}{2}}y^{\frac{5}{2}}
y
3
×
3
5
×
x
2
7
y
2
5
.
Multiply y terms:
Multiply the y terms together using the property of exponents
y
a
×
y
b
=
y
a
+
b
y^{a} \times y^{b} = y^{a+b}
y
a
×
y
b
=
y
a
+
b
:
y
3
×
y
5
2
=
y
3
+
5
2
=
y
11
2
y^{3} \times y^{\frac{5}{2}} = y^{3 + \frac{5}{2}} = y^{\frac{11}{2}}
y
3
×
y
2
5
=
y
3
+
2
5
=
y
2
11
.
Multiply coefficients and combine:
Now, multiply the coefficients (
3
×
5
3 \times \sqrt{5}
3
×
5
) and combine all terms:
3
5
x
7
2
y
11
2
3\sqrt{5}x^{\frac{7}{2}}y^{\frac{11}{2}}
3
5
x
2
7
y
2
11
.
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