Answer :
To find the greatest common factor (GCF) of the expressions
[tex]$$70x^4,\quad 30x^7,\quad 60x^8,$$[/tex]
we follow these steps:
1. Identify the coefficients and the powers of [tex]$x$[/tex]. Here, the coefficients are [tex]$70$[/tex], [tex]$30$[/tex], and [tex]$60$[/tex], and the exponents are [tex]$4$[/tex], [tex]$7$[/tex], and [tex]$8$[/tex], respectively.
2. Find the GCF of the coefficients. We calculate the greatest common divisor (GCD) of [tex]$70$[/tex], [tex]$30$[/tex], and [tex]$60$[/tex]:
- First, find [tex]$\gcd(70, 30) = 10$[/tex].
- Then, find [tex]$\gcd(10, 60) = 10$[/tex].
Therefore, the GCF of the coefficients is [tex]$10$[/tex].
3. Find the common power of [tex]$x$[/tex]. Since [tex]$x$[/tex] appears in each term, we take the smallest exponent among all terms:
[tex]$$\min(4,\, 7,\, 8) = 4.$$[/tex]
Thus, every term includes at least [tex]$x^4$[/tex].
4. Combine the results. The GCF is the product of the GCF of the coefficients and [tex]$x$[/tex] raised to the smallest exponent:
[tex]$$\text{GCF} = 10x^4.$$[/tex]
Hence, the greatest common factor of the expressions is
[tex]$$\boxed{10x^4}.$$[/tex]
[tex]$$70x^4,\quad 30x^7,\quad 60x^8,$$[/tex]
we follow these steps:
1. Identify the coefficients and the powers of [tex]$x$[/tex]. Here, the coefficients are [tex]$70$[/tex], [tex]$30$[/tex], and [tex]$60$[/tex], and the exponents are [tex]$4$[/tex], [tex]$7$[/tex], and [tex]$8$[/tex], respectively.
2. Find the GCF of the coefficients. We calculate the greatest common divisor (GCD) of [tex]$70$[/tex], [tex]$30$[/tex], and [tex]$60$[/tex]:
- First, find [tex]$\gcd(70, 30) = 10$[/tex].
- Then, find [tex]$\gcd(10, 60) = 10$[/tex].
Therefore, the GCF of the coefficients is [tex]$10$[/tex].
3. Find the common power of [tex]$x$[/tex]. Since [tex]$x$[/tex] appears in each term, we take the smallest exponent among all terms:
[tex]$$\min(4,\, 7,\, 8) = 4.$$[/tex]
Thus, every term includes at least [tex]$x^4$[/tex].
4. Combine the results. The GCF is the product of the GCF of the coefficients and [tex]$x$[/tex] raised to the smallest exponent:
[tex]$$\text{GCF} = 10x^4.$$[/tex]
Hence, the greatest common factor of the expressions is
[tex]$$\boxed{10x^4}.$$[/tex]
