Finding the HCF of Algebraic Terms

To find the Highest Common Factor (HCF) of the two terms $28\ p^5q^2$ and $70\ p^3q^4$, we need to find the HCF of the numerical coefficients and the HCF of the variable parts separately.

HCF of Coefficients

First, find the HCF of the numerical coefficients 28 and 70.

• Factors of 28: 1, 2, 4, 7, 14, 28
• Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70
• The highest common factor of 28 and 70 is 14.

HCF of Variables

Next, find the HCF of the variable parts $p^5q^2$ and $p^3q^4$. For each variable, we take the lowest power that appears in both terms.

• For variable $p$: The powers are $p^5$ and $p^3$. The lowest power is $p^3$.
• For variable $q$: The powers are $q^2$ and $q^4$. The lowest power is $q^2$.
• The HCF of the variable parts is $p^3q^2$.

Combining Coefficients and Variables

Finally, combine the HCF of the coefficients and the HCF of the variables to get the overall HCF of the terms.

HCF = (HCF of coefficients) $\times$ (HCF of variables)
HCF = $14 \times p^3q^2$
HCF = $14p^3q^2$

Therefore, the HCF of $28\ p^5q^2$ and $70\ p^3q^4$ is $14\ p^3q^2$. This matches Option 1.