Simplify the Radical Expression fourth root of 8a^3b* fourth root of 10a^2b^7

Solution:

Step:1

Express $10 a^{2} b^{7}$ as $b^{4} \cdot (10 a^{2} b^{3})$.

Step:1.1

Extract $b^{4}$ from the expression: $\sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{10 a^{2} (b^{4} b^{3})}$

Step:1.2

Rearrange to place $a^{2}$ at the front: $\sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{10 b^{4} a^{2} b^{3}}$

Step:1.3

Switch the positions of $10$ and $b^{4}$: $\sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{b^{4} \cdot 10 a^{2} b^{3}}$

Step:1.4

Enclose in brackets: $\sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{b^{4} \cdot 10 (a^{2} b^{3})}$

Step:1.5

Enclose in brackets again for clarity: $\sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{b^{4} \cdot (10 a^{2} b^{3})}$

Step:2

Extract terms from under the radical sign: $\sqrt[4]{8 a^{3} b} \cdot (|b| \sqrt[4]{10 a^{2} b^{3}})$

Step:3

Combine the two radicals: $\sqrt[4]{8 a^{3} b} (|b| \sqrt[4]{10 a^{2} b^{3}})$.

Step:3.1

Apply the product rule for radicals: $\sqrt[4]{10 a^{2} b^{3} (8 a^{3} b)} |b|$

Step:3.2

Multiply $8$ by $10$: $\sqrt[4]{80 a^{2} b^{3} (a^{3} b)} |b|$

Step:3.3

Combine the exponents using the power rule $a^{m} a^{n} = a^{m + n}$: $\sqrt[4]{80 a^{5} b^{3} b} |b|$

Step:3.4

Sum the exponents $3$ and $2$: $\sqrt[4]{80 a^{5} b^{3} b} |b|$

Step:3.5

Raise $b$ to the power of $1$: $\sqrt[4]{80 a^{5} (b^{1} b^{3})} |b|$

Step:3.6

Combine the exponents: $\sqrt[4]{80 a^{5} b^{4}} |b|$

Step:3.7

Sum the exponents $1$ and $3$: $\sqrt[4]{80 a^{5} b^{4}} |b|$

Step:4

Rewrite $80 a^{5} b^{4}$ as $(2 a b)^{4} \cdot (5 a)$.

Step:4.1

Factor $16$ from $80$: $\sqrt[4]{16 (5) a^{5} b^{4}} |b|$

Step:4.2

Represent $16$ as $2^{4}$: $\sqrt[4]{2^{4} \cdot 5 a^{5} b^{4}} |b|$

Step:4.3

Factor out $a^{4}$: $\sqrt[4]{2^{4} \cdot 5 (a^{4} a) b^{4}} |b|$

Step:4.4

Rearrange $a$: $\sqrt[4]{2^{4} \cdot 5 (a^{4}) b^{4} a} |b|$

Step:4.5

Rearrange $5$: $\sqrt[4]{(2^{4} (a^{4})) b^{4} \cdot 5 a} |b|$

Step:4.6

Express $(2^{4} (a^{4})) b^{4}$ as $(2 a b)^{4}$: $\sqrt[4]{(2 a b)^{4} \cdot 5 a} |b|$

Step:4.7

Enclose in brackets: $\sqrt[4]{((2 a b)^{4} \cdot (5 a))} |b|$

Step:5

Extract terms from under the radical: $|2 a b| \sqrt[4]{5 a} |b|$

Step:6

Eliminate non-negative terms from the absolute value: $2 |a b| \sqrt[4]{5 a} |b|$

Step:7

Multiply the terms: $2 |a b| \sqrt[4]{5 a} |b|$.

Step:7.1

To multiply absolute values, multiply the terms inside: $2 |b (a b)| \sqrt[4]{5 a}$

Step:7.2

Raise $b$ to the power of $1$: $2 |a (b^{1} b)| \sqrt[4]{5 a}$

Step:7.3

Raise $b$ to the power of $1$ again: $2 |a (b^{1} b^{1})| \sqrt[4]{5 a}$

Step:7.4

Combine the exponents: $2 |a b^{2}| \sqrt[4]{5 a}$

Step:7.5

Add the exponents $1$ and $1$: $2 |a b^{2}| \sqrt[4]{5 a}$

Step:8

Eliminate non-negative terms from the absolute value: $2 b^{2} |a| \sqrt[4]{5 a}$

Knowledge Notes:

1. Radical Expressions: Expressions that contain roots, such as square roots or fourth roots.

2. Product Rule for Radicals: $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$, which allows us to multiply two radicals with the same index.

3. Power Rule for Exponents: $a^{m} \cdot a^{n} = a^{m + n}$, which is used to combine like bases with exponents.

4. Absolute Value: The absolute value of a number is its distance from zero on the number line, denoted by $|a|$. For non-negative terms, the absolute value is the term itself.

5. Simplifying Radical Expressions: The process involves factoring out perfect powers, combining like terms under the radical, and then simplifying the expression inside and outside the radical.

6. Factoring: The process of breaking down an expression into its constituent factors, which can be useful when simplifying expressions, especially under a radical.

7. Exponent Laws: These laws are used to manipulate expressions with exponents during the simplification process.

8. Rewriting Expressions: Often, expressions can be rewritten in a form that makes it easier to simplify or factor, such as expressing a number as a power of another number.