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Simplify (a) (t-4)/(5t)÷(16-t^(2))/(10t^(2)s),

Simplify\newline(a) t−45t÷16−t210t2s \frac{t-4}{5 t} \div \frac{16-t^{2}}{10 t^{2} s} ,

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Full solution

Q. Simplify\newline(a) t−45t÷16−t210t2s \frac{t-4}{5 t} \div \frac{16-t^{2}}{10 t^{2} s} ,
  1. Write Expression and Identify Operation: Write down the given expression and identify the operation to be performed.\newlineWe are given the expression (t−4)/(5t)÷(16−t2)/(10t2s)(t-4)/(5t) \div (16-t^2)/(10t^2s). We need to divide the first fraction by the second fraction.
  2. Convert Division into Multiplication: Convert the division of fractions into multiplication by the reciprocal of the second fraction.\newlineThe expression (t−4)/(5t)÷(16−t2)/(10t2s)(t-4)/(5t) \div (16-t^2)/(10t^2s) can be rewritten as (t−4)/(5t)×(10t2s)/(16−t2)(t-4)/(5t) \times (10t^2s)/(16-t^2).
  3. Factor Denominator of Second Fraction: Factor the denominator of the second fraction if possible.\newlineThe term 16−t216-t^2 is a difference of squares and can be factored as (4+t)(4−t)(4+t)(4-t).\newlineSo, the expression becomes (t−4)/(5t)×(10t2s)/((4+t)(4−t))(t-4)/(5t) \times (10t^2s)/((4+t)(4-t)).
  4. Simplify by Canceling Common Factors: Simplify the expression by canceling out common factors. We notice that (t−4)(t-4) and (4−t)(4-t) are the same terms with opposite signs, so they cancel each other out as −1-1. We also cancel out tt from the numerator of the first fraction and the denominator of the second fraction. The expression simplifies to −15×10s4+t-\frac{1}{5} \times \frac{10s}{4+t}.
  5. Multiply Numerators and Denominators: Multiply the numerators and denominators of the remaining fractions.\newlineMultiplying the numerators: −1×10s=−10s-1 \times 10s = -10s\newlineMultiplying the denominators: 5×(4+t)=20+5t5 \times (4+t) = 20 + 5t\newlineThe simplified expression is −10s20+5t\frac{-10s}{20 + 5t}.
  6. Check for Further Simplification: Check for any further simplification. The expression −10s/(20+5t)-10s/(20 + 5t) cannot be simplified further as there are no common factors to cancel out.

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