Let's simplify the given expressions step-by-step.
### Part (a): Simplify [tex]\( x \times x \times x \times y \times y \)[/tex]
To simplify this expression, we will group like terms together and use the properties of exponents.
1. Write the expression with grouped like terms:
[tex]\[
x \times x \times x \times y \times y
\][/tex]
2. Group the [tex]\( x \)[/tex] terms together and the [tex]\( y \)[/tex] terms together:
[tex]\[
(x \times x \times x) \times (y \times y)
\][/tex]
3. Use the property of exponents [tex]\( a^m \times a^n = a^{m+n} \)[/tex]:
[tex]\(
x \times x \times x = x^3
\)[/tex]
and
[tex]\(
y \times y = y^2
\)[/tex]
4. Combine the results:
[tex]\[
x^3 \times y^2
\][/tex]
Hence, the simplified form of [tex]\( x \times x \times x \times y \times y \)[/tex] is:
[tex]\[
x^3 y^2
\][/tex]
### Part (b): Simplify [tex]\( 2 \times x \times x \times 3 \times y \times y \times y \)[/tex]
Similarly, we will simplify this expression by grouping like terms together and using properties of exponents and multiplication.
1. Write the expression with grouped like terms:
[tex]\[
2 \times x \times x \times 3 \times y \times y \times y
\][/tex]
2. Group the constants, the [tex]\( x \)[/tex] terms, and the [tex]\( y \)[/tex] terms together:
[tex]\[
(2 \times 3) \times (x \times x) \times (y \times y \times y)
\][/tex]
3. Multiply the constants together:
[tex]\[
2 \times 3 = 6
\][/tex]
4. Use the property of exponents [tex]\( a^m \times a^n = a^{m+n} \)[/tex]:
[tex]\(
x \times x = x^2
\)[/tex]
and
[tex]\(
y \times y \times y = y^3
\)[/tex]
5. Combine the results:
[tex]\[
6 \times x^2 \times y^3
\][/tex]
Hence, the simplified form of [tex]\( 2 \times x \times x \times 3 \times y \times y \times y \)[/tex] is:
[tex]\[
6 x^2 y^3
\][/tex]
The final simplified forms for the expressions are:
a) [tex]\( x^3 y^2 \)[/tex]
b) [tex]\( 6 x^2 y^3 \)[/tex]
