Solve for a 4 = cube root of 8a
The problem provided requires determining the value of the variable a when 'a' raised to the power of 4 is equal to the cube root of 8a. This involves equating a^4 (a to the power of four) with the cube root of the quantity 8a and solving for the unknown variable a.
$4 = \sqrt[3]{8 a}$
Answer
Solution:
Step 1:
Express the given equation in the form $4 = \sqrt[3]{8a}$.
Step 2:
Eliminate the cube root by raising both sides of the equation to the power of three: $(4)^3 = (\sqrt[3]{8a})^3$.
Step 3:
Simplify the equation.
Step 3.1:
Rewrite the cube root as a fractional exponent: $(8a)^{\frac{1}{3}}$.
Step 3.2:
Simplify the left-hand side of the equation.
Step 3.2.1:
Apply the exponent rule to the left-hand side: $((8a)^{\frac{1}{3}})^3$.
Step 3.2.1.1:
Multiply the exponents together: $(8a)^{\frac{1}{3} \cdot 3}$.
Step 3.2.1.1.1:
Use the power of a power rule to simplify the expression: $(8a)^{1}$.
Step 3.2.1.1.2:
Remove the exponent of 1 as it does not change the base: $8a = (4)^3$.
Step 3.3:
Simplify the right-hand side of the equation.
Step 3.3.1:
Calculate $4$ raised to the power of $3$: $8a = 64$.
Step 4:
Solve for $a$ by dividing both sides by $8$.
Step 4.1:
Divide the equation by $8$: $\frac{8a}{8} = \frac{64}{8}$.
Step 4.2:
Simplify the left-hand side.
Step 4.2.1:
Cancel out the common factor of $8$: $a = \frac{64}{8}$.
Step 4.3:
Simplify the right-hand side.
Step 4.3.1:
Divide $64$ by $8$ to find the value of $a$: $a = 8$.
Knowledge Notes:
To solve the equation $4 = \sqrt[3]{8a}$, we follow a systematic process:
Cube Roots and Exponents: The cube root of a number $x$ can be expressed as $x^{\frac{1}{3}}$. When we raise both sides of an equation to the same power, we maintain the equality. This is used to eliminate the cube root.
Algebraic Manipulation: Algebraic manipulation involves simplifying expressions and solving for unknowns using operations such as addition, subtraction, multiplication, division, and exponentiation.
Power Rules: The power of a power rule states that $(a^{m})^{n} = a^{m \cdot n}$. This rule is used to simplify expressions with exponents.
Simplification: Simplification of expressions involves reducing them to their simplest form by performing arithmetic operations and canceling common factors.
Solving Linear Equations: To solve for an unknown in a linear equation, we isolate the variable on one side of the equation using inverse operations. In this case, we divide both sides by the coefficient of the variable to find its value.
