State True or False: `(a^(4))/(b^(3))=(a+a+a+a)/(b+b+b)`

Options

A. True
B. False
C. not sure
D. none of the above

Answer

To determine whether the statement \((\frac{a^{4}}{b^{3}}) = \frac{a+a+a+a}{b+b+b}\) is true or false, we will analyze both sides of the equation step by step.

### Step 1: Analyze the Left-Hand Side (LHS)
The left-hand side of the equation is:
\[ \frac{a^{4}}{b^{3}} \]
This can be expressed as:
\[ \frac{a \cdot a \cdot a \cdot a}{b \cdot b \cdot b} \]
This means we have four \(a\)s multiplied together in the numerator and three \(b\)s multiplied together in the denominator.

### Step 2: Analyze the Right-Hand Side (RHS)
The right-hand side of the equation is:
\[ \frac{a + a + a + a}{b + b + b} \]
Calculating the numerator:
\[ a + a + a + a = 4a \]
And for the denominator:
\[ b + b + b = 3b \]
So, the right-hand side simplifies to:
\[ \frac{4a}{3b} \]

### Step 3: Compare LHS and RHS
Now we compare both sides:
- LHS: \(\frac{a^{4}}{b^{3}}\)
- RHS: \(\frac{4a}{3b}\)

### Step 4: Determine if they are equal
To check if \(\frac{a^{4}}{b^{3}} = \frac{4a}{3b}\), we can cross-multiply:
\[ a^{4} \cdot 3b = 4a \cdot b^{3} \]
This simplifies to:
\[ 3a^{4}b = 4ab^{3} \]
Dividing both sides by \(ab\) (assuming \(a \neq 0\) and \(b \neq 0\)):
\[ 3a^{3} = 4b^{2} \]
This equation does not hold true for all values of \(a\) and \(b\). Therefore, the two sides are not equal.

### Conclusion
Since the left-hand side \(\frac{a^{4}}{b^{3}}\) is not equal to the right-hand side \(\frac{4a}{3b}\), we conclude that the statement is **False**.

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