Find the HCF of \( 24 x^{3} y x^{4} ; 30 x^{2} y^{2} z^{3} \) and \( 36 x^{2} y^{2} z^{3} \)

1. **Express each term in simplified form:** - First term: \(24 x^{3} y x^{4} = 24 x^{3+4} y = 24 x^{7} y\). - Second term: \(30 x^{2} y^{2} z^{3}\) (already simplified). - Third term: \(36 x^{2} y^{2} z^{3}\) (already simplified). 2. **Find the HCF of the numerical coefficients:** - For \(24\): \(2^3 \times 3\). - For \(30\): \(2 \times 3 \times 5\). - For \(36\): \(2^2 \times 3^2\). The common factors are \(2\) (minimum exponent: \(2^1\)) and \(3\) (minimum exponent: \(3^1\)). Thus, the HCF of the coefficients is: \[ 2 \times 3 = 6. \] 3. **Determine the HCF for the variable \(x\):** - Exponents in the terms: \(24x^7y\) has \(x^7\), \(30x^2y^2z^3\) has \(x^2\), and \(36x^2y^2z^3\) has \(x^2\). - Minimum exponent of \(x\) is \(2\). Thus, the contribution from \(x\) is \(x^2\). 4. **Determine the HCF for the variable \(y\):** - Exponents in the terms: \(24x^7y\) has \(y^1\), \(30x^2y^2z^3\) has \(y^2\), and \(36x^2y^2z^3\) has \(y^2\). - Minimum exponent of \(y\) is \(1\). Thus, the contribution from \(y\) is \(y\). 5. **Determine the HCF for the variable \(z\):** - \(z\) appears in the second and third terms with exponent \(3\), but is absent in the first term (i.e., exponent \(0\)). - Minimum exponent of \(z\) is \(0\) (meaning \(z\) is not a common factor). 6. **Multiply the common factors together:** \[ \text{HCF} = 6 \times x^2 \times y = 6x^2y. \] Thus, the highest common factor is \(6x^2y\).