Question

using the distributive property, an expression equivalent to 48 + 30 is
\\( 5(6 + 8) \\).
\\( (6)(8) + (3)(10) \\).
\\( (3)(16) + (6)(5) \\).
\\( 6(8 + 5) \\).

Explanation:

Step1: Find GCD of 48 and 30

The greatest common divisor (GCD) of 48 and 30. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. GCD is 6.

Step2: Apply distributive property

The distributive property is \(a(b + c)=ab + ac\). We can factor out 6 from 48 and 30. \(48=6\times8\), \(30 = 6\times5\). So \(48 + 30=6\times8+6\times5=6(8 + 5)\).

Step3: Check other options

• For \(5(6 + 8)\): \(5\times6+5\times8 = 30+40=70

eq48 + 30 = 78\).

• For \((6)(8)+(3)(10)\): \(48+30 = 78\)? Wait, \((6)(8)=48\), \((3)(10)=30\), but this is not using distributive property correctly as we need a common factor. Wait no, \(48+30\), if we factor 3, \(3\times16+3\times10=3(16 + 10)\), but the option is \((6)(8)+(3)(10)\) which is \(48 + 30\) but not a single factored form. Wait our first step: \(48+30 = 6\times8+6\times5=6(8 + 5)\) which matches the last option. Let's recalculate:
• Option 1: \(5(6 + 8)=5\times14 = 70

eq78\)

• Option 2: \((6)(8)+(3)(10)=48 + 30=78\) but it's not a factored form using distributive property (it's just splitting into two products, not factoring out a common factor)
• Option 3: \((3)(16)+(6)(5)=48+30 = 78\), again not a factored form with a single common factor.
• Option 4: \(6(8 + 5)=6\times13 = 78\), and \(48+30 = 78\), and this is using distributive property \(a(b + c)=ab+ac\) with \(a = 6\), \(b = 8\), \(c = 5\) (since \(6\times8=48\), \(6\times5 = 30\))

Answer:

\(6(8 + 5)\) (the last option, which is the fourth option in the list: \(6(8 + 5)\))