Simplify square root of 2a* square root of 14a^3* square root of 5a
The question asks for the simplification of a mathematical expression that involves the multiplication of three radical terms (also known as square roots). Each term under the square root contains variables and coefficients. The process of simplification typically involves multiplying the terms under the square root to combine them into a single radical. After the multiplication, it is expected to further simplify the expression by factoring out perfect squares if possible and reducing the expression to its simplest radical form. In this case, the expression consists of the square root of 2a, the square root of 14a^3, and the square root of 5a, all multiplied together.
$\sqrt{2 a \cdot \sqrt{14 a^{3}} \cdot \sqrt{5 a}}$
Answer
Solution:
Step:1
Apply the product rule for radicals to combine them: $\sqrt{2a} \cdot \sqrt{14a^3} \cdot \sqrt{5a}$.
Step:2
Calculate the product of $14$ and $5$: $\sqrt{2a \cdot \sqrt{70a^4}}$.
Step:3
Combine like terms by adding exponents of $a$.
Step:3.1
Rearrange to place $a^3$ next to $a$: $\sqrt{2a \cdot \sqrt{70a^3a}}$.
Step:3.2
Combine $a^3$ and $a$.
Step:3.2.1
Express $a$ as $a^1$: $\sqrt{2a \cdot \sqrt{70a^3a^1}}$.
Step:3.2.2
Apply the exponent rule $a^m \cdot a^n = a^{m+n}$: $\sqrt{2a \cdot \sqrt{70a^{4}}}$.
Step:3.3
Sum the exponents $3$ and $1$: $\sqrt{2a \cdot \sqrt{70a^4}}$.
Step:4
Express $70a^4$ as $(a^2)^2 \cdot 70$.
Step:4.1
Rewrite $a^4$ as $(a^2)^2$: $\sqrt{2a \cdot \sqrt{70(a^2)^2}}$.
Step:4.2
Switch the order of $70$ and $(a^2)^2$: $\sqrt{2a \cdot \sqrt{(a^2)^2 \cdot 70}}$.
Step:5
Extract terms from under the radical sign: $\sqrt{2a \cdot a^2 \cdot \sqrt{70}}$.
Step:6
Multiply $a$ by $a^2$ by adding their exponents.
Step:6.1
Place $a^2$ next to $a$: $\sqrt{2a^2a \cdot \sqrt{70}}$.
Step:6.2
Multiply $a^2$ by $a$.
Step:6.2.1
Express $a$ as $a^1$: $\sqrt{2a^2a^1 \cdot \sqrt{70}}$.
Step:6.2.2
Combine exponents using the rule $a^m \cdot a^n = a^{m+n}$: $\sqrt{2a^{3} \cdot \sqrt{70}}$.
Step:6.3
Add the exponents $2$ and $1$: $\sqrt{2a^3 \cdot \sqrt{70}}$.
Step:7
Rearrange $2a^3 \cdot \sqrt{70}$ as $a^2 \cdot (2a \cdot \sqrt{70})$.
Step:7.1
Factor out $a^2$: $\sqrt{a^2 \cdot 2a \cdot \sqrt{70}}$.
Step:7.2
Switch the positions of $2$ and $a^2$: $\sqrt{a^2 \cdot 2a \cdot \sqrt{70}}$.
Step:7.3
Enclose $2a \cdot \sqrt{70}$ in parentheses: $\sqrt{a^2 \cdot (2a \cdot \sqrt{70})}$.
Step:7.4
Enclose $2a \cdot \sqrt{70}$ in parentheses for clarity: $\sqrt{a^2 \cdot (2a \cdot \sqrt{70})}$.
Step:8
Extract terms from under the radical, simplifying to: $a \cdot \sqrt{2a \cdot \sqrt{70}}$.
Knowledge Notes:
The problem involves simplifying a product of radical expressions. The key knowledge points and rules used in the solution are:
Product Rule for Radicals: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. This rule allows us to combine or separate radicals that are being multiplied.
Exponent Rules:
Power Rule: $(a^m)^n = a^{mn}$.
Product of Powers Rule: $a^m \cdot a^n = a^{m+n}$.
These rules help us to manipulate expressions with exponents, particularly when combining like bases.
Simplifying Radicals: When a term inside a radical can be expressed as a perfect square (or a higher power that matches the index of the radical), it can be taken out of the radical. For example, $\sqrt{a^2} = a$.
Rearranging Terms: Sometimes, it's necessary to rearrange terms to make it easier to apply the product rule for radicals or the exponent rules.
Combining Like Terms: When terms have the same base and are multiplied together, we can add their exponents to combine them into a single term.
By applying these rules and knowledge points step by step, the original expression is simplified to its most reduced form.
