Photo AI

4 (a) Show that $a^5 \times (a^3)^2$ can be expressed as $a^{11}$ - OCR - GCSE Maths - Question 4 - 2018 - Paper 1

Question icon

Question 4

4-(a)-Show-that-$a^5-\times-(a^3)^2$-can-be-expressed-as-$a^{11}$-OCR-GCSE Maths-Question 4-2018-Paper 1.png

4 (a) Show that $a^5 \times (a^3)^2$ can be expressed as $a^{11}$. (b) Write $\frac{1}{125} \times 25^5$ as a power of 5.

Worked Solution & Example Answer:4 (a) Show that $a^5 \times (a^3)^2$ can be expressed as $a^{11}$ - OCR - GCSE Maths - Question 4 - 2018 - Paper 1

Step 1

Show that $a^5 \times (a^3)^2$ can be expressed as $a^{11}$

96%

114 rated

Answer

To show that a5×(a3)2a^5 \times (a^3)^2 can be expressed as a11a^{11}, we will first apply the power rule of exponents. The power rule states that (xm)n=xm⋅n(x^m)^n = x^{m \cdot n}.

  1. Calculate (a3)2(a^3)^2:
    (a3)2=a3⋅2=a6.(a^3)^2 = a^{3 \cdot 2} = a^6.

  2. Now substitute this back into the original expression:
    a5×a6.a^5 \times a^6.

  3. Using the product rule for exponents, which states that xm×xn=xm+nx^m \times x^n = x^{m+n}:
    a5×a6=a5+6=a11.a^5 \times a^6 = a^{5+6} = a^{11}.

Thus, we have shown that a5×(a3)2a^5 \times (a^3)^2 can be expressed as a11a^{11}.

Step 2

Write $\frac{1}{125} \times 25^5$ as a power of 5

99%

104 rated

Answer

To write 1125×255\frac{1}{125} \times 25^5 as a power of 5, we first convert both 125 and 25 to their respective powers of 5:

  1. Recognizing the powers:
    125=53125 = 5^3
    and
    25=52.25 = 5^2.

  2. Thus, we can rewrite the expression:
    1125=5−3.\frac{1}{125} = 5^{-3}.
    Therefore,
    1125×255=5−3×(52)5.\frac{1}{125} \times 25^5 = 5^{-3} \times (5^2)^5.

  3. Now, applying the power rule again to (52)5(5^2)^5:
    (52)5=52⋅5=510.(5^2)^5 = 5^{2 \cdot 5} = 5^{10}.

  4. Combining the powers:
    5−3×510=5−3+10=57.5^{-3} \times 5^{10} = 5^{-3 + 10} = 5^{7}.

Therefore, we have expressed 1125×255\frac{1}{125} \times 25^5 as 575^7.

Join the GCSE students using SimpleStudy...

97% of Students

Report Improved Results

98% of Students

Recommend to friends

100,000+

Students Supported

1 Million+

Questions answered

;