Certainly, let us simplify each expression step-by-step.
### Problem 1:
Simplify [tex]\(2a^4 + 2x^4\)[/tex].
Step-by-Step Solution:
We factor out the greatest common factor (GCF) from the terms.
1. The GCF of [tex]\(2a^4\)[/tex] and [tex]\(2x^4\)[/tex] is 2.
2. Factor 2 out of each term:
[tex]\[
2a^4 + 2x^4 = 2(a^4 + x^4)
\][/tex]
Hence, the simplified form is:
[tex]\[
2(a^4 + x^4)
\][/tex]
### Problem 2:
Simplify [tex]\(3d^6(d - 2)\)[/tex].
Step-by-Step Solution:
1. Distribute [tex]\(3d^6\)[/tex] into each term inside the parentheses:
[tex]\[
3d^6(d - 2) = 3d^6 \cdot d - 3d^6 \cdot 2
\][/tex]
2. Simplify each term:
[tex]\[
3d^6 \cdot d = 3d^{7}
\][/tex]
[tex]\[
3d^6 \cdot 2 = 6d^6
\][/tex]
3. Combine the results:
[tex]\[
3d^7 - 6d^6
\][/tex]
Hence, the simplified form is:
[tex]\[
3d^7 - 6d^6
\][/tex]
### Problem 3:
Simplify [tex]\(\sqrt{4} \times 2\)[/tex].
Step-by-Step Solution:
1. Calculate the square root of 4:
[tex]\[
\sqrt{4} = 2
\][/tex]
2. Multiply the result by 2:
[tex]\[
2 \times 2 = 4
\][/tex]
Hence, the simplified form is:
[tex]\[
4
\][/tex]
### Problem 4:
Simplify [tex]\(\sqrt[3]{8y^3}\)[/tex].
Step-by-Step Solution:
1. Utilize properties of exponents:
[tex]\[
\sqrt[3]{8y^3} = (8y^3)^{\frac{1}{3}}
\][/tex]
2. Recall that [tex]\(8 = 2^3\)[/tex], so:
[tex]\[
(8y^3)^{\frac{1}{3}} = (2^3 y^3)^{\frac{1}{3}}
\][/tex]
3. Apply the power of a power property:
[tex]\[
(2^3 y^3)^{\frac{1}{3}} = 2^{\frac{3}{3}} y^{\frac{3}{3}}
\][/tex]
[tex]\[
= 2^1 y^1 = 2y
\][/tex]
Hence, the simplified form is:
[tex]\[
2y
\][/tex]
### Problem 5:
Simplify [tex]\(81a^6 + \sqrt[3]{27a^9}\)[/tex].
Step-by-Step Solution:
1. Simplify [tex]\( \sqrt[3]{27a^9} \)[/tex]:
[tex]\[
27a^9 = (3^3 a^9) = (3^3 (a^3)^3)
\][/tex]
[tex]\[
\sqrt[3]{27a^9} = (3^3 (a^3)^3)^{\frac{1}{3}} = 3(a^3)^1 = 3a^3
\][/tex]
2. Combine the terms:
[tex]\[
81a^6 + 3a^3
\][/tex]
Hence, the simplified form is:
[tex]\[
81a^6 + 3a^3
\][/tex]
### Problem 6:
Simplify [tex]\(2x^4 - 2x^2\)[/tex].
Step-by-Step Solution:
1. Factor out the greatest common factor (GCF) from the terms:
[tex]\[
2x^4 - 2x^2 = 2x^2(x^2 - 1)
\][/tex]
2. Recognize that [tex]\(x^2 - 1\)[/tex] is a difference of squares:
[tex]\[
x^2 - 1 = (x + 1)(x - 1)
\][/tex]
So,
[tex]\[
2x^2(x^2 - 1) = 2x^2(x + 1)(x - 1)
\][/tex]
Hence, the simplified form is:
[tex]\[
2x^2(x + 1)(x - 1)
\][/tex]
These steps offer a clear and thorough process for each problem, ensuring a deep understanding of the simplification methods involved.
