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Product Rule
Reference
> Algebra: Exponents
Description
The Product Rule states that:
x
a
x
b
=
x
a
+
b
{x}^{a}{x}^{b}={x}^{a+b}
x
a
x
b
=
x
a
+
b
Examples
Example 1
x
3
x
2
x
4
{x}^{3}{x}^{2}{x}^{4}
x
3
x
2
x
4
1
Use
Product Rule
:
x
a
x
b
=
x
a
+
b
{x}^{a}{x}^{b}={x}^{a+b}
x
a
x
b
=
x
a
+
b
.
x
3
+
2
+
4
{x}^{3+2+4}
x
3
+
2
+
4
2
Simplify
3
+
2
3+2
3
+
2
to
5
5
5
.
x
5
+
4
{x}^{5+4}
x
5
+
4
3
Simplify
5
+
4
5+4
5
+
4
to
9
9
9
.
x
9
{x}^{9}
x
9
Done
x^9
Example 2
(
x
+
3
)
2
(
x
+
3
)
3
{(x+3)}^{2}{(x+3)}^{3}
(
x
+
3
)
2
(
x
+
3
)
3
1
Use
Product Rule
:
x
a
x
b
=
x
a
+
b
{x}^{a}{x}^{b}={x}^{a+b}
x
a
x
b
=
x
a
+
b
.
(
x
+
3
)
5
{(x+3)}^{5}
(
x
+
3
)
5
Done
(x+3)^5
Example 3
sin
3
x
cos
x
sin
x
\sin^{3}x\cos{x}\sin{x}
sin
3
x
cos
x
sin
x
1
Use
Product Rule
:
x
a
x
b
=
x
a
+
b
{x}^{a}{x}^{b}={x}^{a+b}
x
a
x
b
=
x
a
+
b
.
sin
4
x
cos
x
\sin^{4}x\cos{x}
sin
4
x
cos
x
Done
sin(x)^4*cos(x)
English
Product Rule