Solve for d 6 = square root of d/16
This is an algebraic problem where you are asked to find the value of the variable 'd'. The equation presented is an equality involving the square root function, where 6 is equal to the square root of 'd' divided by 16. Your task is to manipulate the equation algebraically to isolate 'd' and solve for its numerical value.
$6 = \sqrt{\frac{d}{16}}$
Answer
Solution:
Step 1:
Express the equation in the form $\sqrt{\frac{d}{16}} = 6$.
Step 2:
Eliminate the square root by squaring both sides of the equation to get $\left(\sqrt{\frac{d}{16}}\right)^2 = 6^2$.
Step 3:
Perform simplification on both sides.
Step 3.1:
Rewrite the square root as a power: $\left(\frac{d}{16}\right)^{\frac{1}{2}}$.
Step 3.2:
Simplify the left-hand side.
Step 3.2.1:
Apply the exponent rule to the left-hand side: $\left(\frac{d}{16}\right)^{\frac{1}{2} \cdot 2}$.
Step 3.2.1.1:
Combine the exponents by multiplying them: $\left(\frac{d}{16}\right)^{1}$.
Step 3.2.1.2:
The expression simplifies to $\frac{d}{16} = 6^2$.
Step 3.3:
Simplify the right-hand side to get $\frac{d}{16} = 36$.
Step 4:
Isolate $d$ by solving the equation.
Step 4.1:
Multiply both sides by $16$ to get $16 \cdot \frac{d}{16} = 16 \cdot 36$.
Step 4.2:
Simplify both sides.
Step 4.2.1:
On the left, the $16$s cancel out, leaving $d = 16 \cdot 36$.
Step 4.2.2:
Calculate the product on the right to find $d = 576$.
Knowledge Notes:
The problem involves solving an equation with a square root. Here are the relevant knowledge points:
Square Roots: The square root of a number $x$ is a value that, when multiplied by itself, gives $x$. The square root of $\frac{d}{16}$ is written as $\sqrt{\frac{d}{16}}$.
Squaring Both Sides: To eliminate a square root, you can square both sides of the equation. This must be done carefully to ensure the integrity of the equation is maintained.
Exponent Rules: When you have an exponent raised to another exponent, you multiply the exponents. This is shown by the formula $(a^m)^n = a^{mn}$.
Simplification: After squaring both sides, it is important to simplify the equation by performing any possible arithmetic operations or canceling out common factors.
Isolating the Variable: The last step in solving an equation is to isolate the variable you are solving for. In this case, by multiplying both sides by $16$, we isolate $d$.
Multiplication: The final step involves basic multiplication to solve for the value of $d$ after it has been isolated.
By understanding these concepts, you can solve a wide range of algebraic equations involving square roots and other exponents.
