Solve for r cube root of 7r+5=-3

Solution:

Step 1

Cube both sides to eliminate the cube root:

$(\sqrt[3]{7r+5}{)}^3=(-3{)}^3$

Step 2

Expand and simplify both sides of the equation:

Step 2.1

Express the cube root as a fractional exponent:

$(7r+5{)}^{\frac{1}{3}}$

Step 2.2

Raise both sides to the power of 3:

Step 2.2.1

Apply the exponentiation:

$(7r+5{)}^{\frac{1}{3}\cdot 3}=(-3{)}^3$

Step 2.2.1.1

Use the power of a power rule:

$(7r+5{)}^{\frac{1}{3}\cdot 3}=(-3{)}^3$

Step 2.2.1.1.1

Simplify the left side by canceling out the exponents:

$(7r+5{)}^1=(-3{)}^3$

Step 2.2.1.1.2

The expression simplifies to:

$7r+5=(-3{)}^3$

Step 2.3

Cube the number -3:

$7r+5=-27$

Step 3

Isolate the variable $r$:

Step 3.1

Subtract 5 from both sides:

$7r=-27-5$

Step 3.2

Divide by 7 to solve for $r$:

Step 3.2.1

Divide both sides by 7:

$\frac{7r}{7}=\frac{-32}{7}$

Step 3.2.2

Simplify the equation:

$r=\frac{-32}{7}$

Step 3.2.3

Write the negative sign in front of the fraction:

$r=-\frac{32}{7}$

Step 4

Present the solution in different forms:

Exact Form: $r=-\frac{32}{7}$ Decimal Form: $r\approx -4.5714$ Mixed Number Form: $r=-4\frac{4}{7}$

Knowledge Notes:

1. Cube Root: The cube root of a number $a$ is a number $b$ such that $b^3=a$. It is denoted as $\sqrt[3]{a}$.

2. Exponentiation: Raising a number to a power is a mathematical operation, written as $a^n$, involving two numbers, the base $a$ and the exponent $n$.

3. Fractional Exponents: A fractional exponent represents both a root and a power. For example, $a^{\frac{1}{n}}$ is the nth root of $a$.

4. Power of a Power Rule: For any nonzero number $a$ and any integers $m$ and $n$, the rule states that $(a^m{)}^n=a^{mn}$.

5. Simplifying Equations: Involves combining like terms, canceling common factors, and performing arithmetic operations to find the value of a variable.

6. Negative Exponents: A negative exponent represents the reciprocal of the base raised to the absolute value of the exponent, e.g., $a^{-n}=\frac{1}{a^n}$.

7. Exact, Decimal, and Mixed Number Forms: Solutions can be expressed exactly as fractions, approximately as decimals, or as mixed numbers combining whole numbers and fractions.